Corresponding and alternate angles are equal

4.G.A.1 · adapt · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

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 the transversals meet 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 that same intersection.

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Understand

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 that opens downward toward q is marked with a circle. I need that circle angle.

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 circle is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the circle angle at the crossing point.

Constraints

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

Plan

#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 circle angle), 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.

Execute

#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 circle is the apex angle.

\angle(\text{circle}) = 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.

#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 55 deg.

#7 Identify Subproblems 4.MD.C.7

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

\angle(\text{circle}) = 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

Review

The base angles 55 deg and 40 deg add to 95 deg, leaving 85 deg for the apex, which is just under a right angle and matches the moderately sharp crossing shown. All three angles 55 + 40 + 85 = 180 deg.

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 circle is its straight-line partner, 180 - 95 = 85 deg.

Standards · min 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.

💡 Find the hidden triangle, slide the 55 deg down between the parallel lines, and the last angle is just what is left of 180 deg!