← 4-1 · Quadrilateral angles sum to 360 degrees · Angle Facts in a Figure

Quadrilateral angles sum to 360 degrees · 8 practice problems

4.MD.C.74.G.A.1

Generated variants — 8

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

Variant 1 answer: 50 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $90^\circ$ , the left angle is $95^\circ$ , the right angle is $85^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $40^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 90, 95, and 85 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 40-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 90, left 95, right 85, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 40 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 90^\circ - 95^\circ - 85^\circ = 90^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 40 degrees, b = 90 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 40^\circ - 90^\circ = 50^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 50 degrees

Review

Check the quadrilateral: 90 + 95 + 85 + 90 = 360 degrees. Check the line: 40 + 90 + 50 = 180 degrees. Both totals are exact, so a = 50 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 40 and b is 180 - 40 - 90, which directly gives the leftover 50 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 2 answer: 55 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $95^\circ$ , the left angle is $100^\circ$ , the right angle is $75^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $35^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 95, 100, and 75 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 35-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 95, left 100, right 75, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 35 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 95^\circ - 100^\circ - 75^\circ = 90^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 35 degrees, b = 90 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 35^\circ - 90^\circ = 55^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 55 degrees

Review

Check the quadrilateral: 95 + 100 + 75 + 90 = 360 degrees. Check the line: 35 + 90 + 55 = 180 degrees. Both totals are exact, so a = 55 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 35 and b is 180 - 35 - 90, which directly gives the leftover 55 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 3 answer: 60 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $110^\circ$ , the left angle is $70^\circ$ , the right angle is $90^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $30^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 110, 70, and 90 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 30-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 110, left 70, right 90, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 30 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 110^\circ - 70^\circ - 90^\circ = 90^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 30 degrees, b = 90 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 30^\circ - 90^\circ = 60^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 60 degrees

Review

Check the quadrilateral: 110 + 70 + 90 + 90 = 360 degrees. Check the line: 30 + 90 + 60 = 180 degrees. Both totals are exact, so a = 60 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 30 and b is 180 - 30 - 90, which directly gives the leftover 60 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 4 answer: 50 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $100^\circ$ , the left angle is $90^\circ$ , the right angle is $80^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $40^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 100, 90, and 80 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 40-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 100, left 90, right 80, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 40 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 100^\circ - 90^\circ - 80^\circ = 90^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 40 degrees, b = 90 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 40^\circ - 90^\circ = 50^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 50 degrees

Review

Check the quadrilateral: 100 + 90 + 80 + 90 = 360 degrees. Check the line: 40 + 90 + 50 = 180 degrees. Both totals are exact, so a = 50 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 40 and b is 180 - 40 - 90, which directly gives the leftover 50 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 5 answer: 30 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $120^\circ$ , the left angle is $60^\circ$ , the right angle is $80^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $50^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 120, 60, and 80 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 50-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 120, left 60, right 80, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 50 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 120^\circ - 60^\circ - 80^\circ = 100^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 50 degrees, b = 100 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 50^\circ - 100^\circ = 30^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 30 degrees

Review

Check the quadrilateral: 120 + 60 + 80 + 100 = 360 degrees. Check the line: 50 + 100 + 30 = 180 degrees. Both totals are exact, so a = 30 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 50 and b is 180 - 50 - 100, which directly gives the leftover 30 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 6 answer: 70 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $130^\circ$ , the left angle is $70^\circ$ , the right angle is $70^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $20^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 130, 70, and 70 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 20-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 130, left 70, right 70, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 20 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 130^\circ - 70^\circ - 70^\circ = 90^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 20 degrees, b = 90 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 20^\circ - 90^\circ = 70^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 70 degrees

Review

Check the quadrilateral: 130 + 70 + 70 + 90 = 360 degrees. Check the line: 20 + 90 + 70 = 180 degrees. Both totals are exact, so a = 70 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 20 and b is 180 - 20 - 90, which directly gives the leftover 70 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 7 answer: 25 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $100^\circ$ , the left angle is $80^\circ$ , the right angle is $70^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $45^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 100, 80, and 70 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 45-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 100, left 80, right 70, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 45 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 100^\circ - 80^\circ - 70^\circ = 110^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 45 degrees, b = 110 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 45^\circ - 110^\circ = 25^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 25 degrees

Review

Check the quadrilateral: 100 + 80 + 70 + 110 = 360 degrees. Check the line: 45 + 110 + 25 = 180 degrees. Both totals are exact, so a = 25 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 45 and b is 180 - 45 - 110, which directly gives the leftover 25 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!

Variant 8 answer: 75 degrees

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $85^\circ$ , the left angle is $105^\circ$ , the right angle is $90^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line -- from left to right -- are a $25^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

Show solution

Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 85, 105, and 90 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 25-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 85, left 105, right 90, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 25 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

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

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract the line-left angle and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 85^\circ - 105^\circ - 90^\circ = 80^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 25 degrees, b = 80 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

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

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 75 degrees

Review

Check the quadrilateral: 85 + 105 + 90 + 80 = 360 degrees. Check the line: 25 + 80 + 75 = 180 degrees. Both totals are exact, so a = 75 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 25 and b is 180 - 25 - 80, which directly gives the leftover 75 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!