← Corresponding and alternate angles are equal · Angle Facts in a Figure

Corresponding and alternate angles are equal · 10 practice problems

4.G.A.1

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 100 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $40^\circ$ . Where they meet the lower line $q$ , one angle is marked as $40^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 40 deg; one angle where a transversal meets the bottom line q is 40 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 40 deg.
A base angle made with line q is 40 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 40 deg is one base angle directly. The 40 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 40 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 40 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 40 deg.

40^\circ \text{ at } p = 40^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 40 deg and 40 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 40^\circ - 40^\circ = 100^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 100 degrees

4 · Reviewdoes it hold up?

The base angles 40 deg and 40 deg add to 80 deg, leaving 100 deg for the apex. All three angles 40 + 40 + 100 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 40 + 40 = 80 deg; the angle a is its straight-line partner, 180 - 80 = 100 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 40 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 40 and 40 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 2 easy answer: 100 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $45^\circ$ . Where they meet the lower line $q$ , one angle is marked as $35^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 45 deg; one angle where a transversal meets the bottom line q is 35 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 45 deg.
A base angle made with line q is 35 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 35 deg is one base angle directly. The 45 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 45 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 45 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 45 deg.

45^\circ \text{ at } p = 45^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 45 deg and 35 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 45^\circ - 35^\circ = 100^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 100 degrees

4 · Reviewdoes it hold up?

The base angles 45 deg and 35 deg add to 80 deg, leaving 100 deg for the apex. All three angles 45 + 35 + 100 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 45 + 35 = 80 deg; the angle a is its straight-line partner, 180 - 80 = 100 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 45 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 45 and 35 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 3 easy answer: 85 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $55^\circ$ . Where they meet the lower line $q$ , one angle is marked as $40^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 55 deg; one angle where a transversal meets the bottom line q is 40 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 55 deg.
A base angle made with line q is 40 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 40 deg is one base angle directly. The 55 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 55 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 55 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 55 deg.

55^\circ \text{ at } p = 55^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 55 deg and 40 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 55^\circ - 40^\circ = 85^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 85 degrees

4 · Reviewdoes it hold up?

The base angles 55 deg and 40 deg add to 95 deg, leaving 85 deg for the apex. All three angles 55 + 40 + 85 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 55 + 40 = 95 deg; the angle a is its straight-line partner, 180 - 95 = 85 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 55 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 55 and 40 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 4 medium answer: 70 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $50^\circ$ . Where they meet the lower line $q$ , one angle is marked as $60^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 50 deg; one angle where a transversal meets the bottom line q is 60 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 50 deg.
A base angle made with line q is 60 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 60 deg is one base angle directly. The 50 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 50 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 50 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 50 deg.

50^\circ \text{ at } p = 50^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 50 deg and 60 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 50^\circ - 60^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 50 deg and 60 deg add to 110 deg, leaving 70 deg for the apex. All three angles 50 + 60 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 50 + 60 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 50 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 50 and 60 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 5 medium answer: 70 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $60^\circ$ . Where they meet the lower line $q$ , one angle is marked as $50^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 60 deg; one angle where a transversal meets the bottom line q is 50 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 60 deg.
A base angle made with line q is 50 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 50 deg is one base angle directly. The 60 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 60 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 60 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 60 deg.

60^\circ \text{ at } p = 60^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 60 deg and 50 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 60^\circ - 50^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 60 deg and 50 deg add to 110 deg, leaving 70 deg for the apex. All three angles 60 + 50 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 60 + 50 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 60 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 60 and 50 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 6 medium answer: 70 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $65^\circ$ . Where they meet the lower line $q$ , one angle is marked as $45^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 65 deg; one angle where a transversal meets the bottom line q is 45 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 65 deg.
A base angle made with line q is 45 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 45 deg is one base angle directly. The 65 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 65 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 65 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 65 deg.

65^\circ \text{ at } p = 65^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 65 deg and 45 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 65^\circ - 45^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 65 deg and 45 deg add to 110 deg, leaving 70 deg for the apex. All three angles 65 + 45 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 65 + 45 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 65 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 65 and 45 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 7 medium answer: 80 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $35^\circ$ . Where they meet the lower line $q$ , one angle is marked as $65^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 35 deg; one angle where a transversal meets the bottom line q is 65 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 35 deg.
A base angle made with line q is 65 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 65 deg is one base angle directly. The 35 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 35 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 35 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 35 deg.

35^\circ \text{ at } p = 35^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 35 deg and 65 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 35^\circ - 65^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 35 deg and 65 deg add to 100 deg, leaving 80 deg for the apex. All three angles 35 + 65 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 35 + 65 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 35 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 35 and 65 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 8 hard answer: 80 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $70^\circ$ . Where they meet the lower line $q$ , one angle is marked as $30^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 70 deg; one angle where a transversal meets the bottom line q is 30 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 70 deg.
A base angle made with line q is 30 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 30 deg is one base angle directly. The 70 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 70 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 70 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 70 deg.

70^\circ \text{ at } p = 70^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 70 deg and 30 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 70^\circ - 30^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 70 deg and 30 deg add to 100 deg, leaving 80 deg for the apex. All three angles 70 + 30 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 70 + 30 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 70 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 70 and 30 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 9 hard answer: 80 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $25^\circ$ . Where they meet the lower line $q$ , one angle is marked as $75^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 25 deg; one angle where a transversal meets the bottom line q is 75 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 25 deg.
A base angle made with line q is 75 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 75 deg is one base angle directly. The 25 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 25 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 25 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 25 deg.

25^\circ \text{ at } p = 25^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 25 deg and 75 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 25^\circ - 75^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 25 deg and 75 deg add to 100 deg, leaving 80 deg for the apex. All three angles 25 + 75 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 25 + 75 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 25 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 25 and 75 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 10 hard answer: 75 degrees

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $80^\circ$ . Where they meet the lower line $q$ , one angle is marked as $25^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 80 deg; one angle where a transversal meets the bottom line q is 25 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 80 deg.
A base angle made with line q is 25 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 25 deg is one base angle directly. The 80 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 80 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 80 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 80 deg.

80^\circ \text{ at } p = 80^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 80 deg and 25 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 80^\circ - 25^\circ = 75^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 75 degrees

4 · Reviewdoes it hold up?

The base angles 80 deg and 25 deg add to 105 deg, leaving 75 deg for the apex. All three angles 80 + 25 + 75 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 80 + 25 = 105 deg; the angle a is its straight-line partner, 180 - 105 = 75 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 80 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 80 and 25 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!