A straight line is 180 degrees

4.MD.C.7 · take · grade 4

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

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $50^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $50^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $50^\circ$ angle (on the other side of the line) is angle $a$ .

Show solution

Understand

Three straight lines cross at one point. Along one line, from left to right, there is a right angle (90 degrees), angle b, and a 50-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 50-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 50 degrees add to 180 degrees.
Angle a is across the crossing point from the 50-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

Plan

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

Subproblem 1: find b by subtracting the 90 and 50 from the 180-degree straight line. Subproblem 2: angle a is opposite the 50-degree angle across the crossing point, so it equals 50 degrees.

Execute

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 50 degrees make up the 180-degree straight angle. Subtract the two known angles.

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

The flat line totals 180 degrees; the middle piece is what is left after the right angle and the 50-degree angle.

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 50-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 50 degrees. (You can also see it because a and 50 each complete the same straight line with the same partner angle.)

a = 50^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 50 degrees, angle b = 40 degrees

Review

Check the line: 90 + 40 + 50 = 180 degrees. And the vertical angle to 50 degrees must also be 50 degrees, so a = 50 degrees is consistent with the crossing.

Use the same straight-line idea for a (tool 7): along the line carrying the 50-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 40, so a = 180 - 130 = 50 degrees, matching the vertical-angle result.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡 A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!