Adjacent angles of a parallelogram sum to 180

4.G.A.2 · adapt · grade 4

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

Quadrilateral ABCD is a parallelogram. Segment AM and segment AD have the same length. Find the measure of angle ⓐ.

Figure description: Parallelogram ABCD lies tilted to one side (A top-left, D top-right, B bottom-left, C bottom-right). A point M lies below the bottom side BC, and segments are drawn from vertex A to point M and to vertex D so that segment AM equals segment AD in length. The angle marked at point M is $40^\circ$ , and the angle marked near vertex D is $20^\circ$ . The angle to find, ⓐ, is the one marked at vertex A.

Show solution

Understand

ABCD is a parallelogram (A top-left, D top-right, B bottom-left, C bottom-right). A point M lies below side BC, and segments AM and AD are drawn with AM = AD. The angle at M (angle AMD) is 40 deg and the angle near D (angle MDC, between DM and side DC) is 20 deg. I need angle a at vertex A, which is angle DAM.

Givens

ABCD is a parallelogram.
Segment AM equals segment AD (triangle AMD is isosceles with apex A).
Angle AMD at M is 40 deg.
Angle MDC at D is 20 deg.
M is below side BC.

Unknowns

The measure of angle a = angle DAM at vertex A.

Constraints

In an isosceles triangle the two base angles (opposite the equal sides) are equal.
The three angles of a triangle add to 180 deg.

Plan

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

Focus on triangle AMD. Since AM = AD, it is isosceles, so the base angles at M and at D inside this triangle are equal. The given 40 deg at M is one base angle, which forces the matching base angle at D, and then the apex angle a at A is whatever is left to make the triangle's angles total 180 deg.

Execute

#1 Draw a Diagram 4.G.A.2

Because AM = AD, triangle AMD is isosceles with apex A, so its two base angles, angle AMD (at M) and angle ADM (at D), are equal.

\angle ADM = \angle AMD

Two equal sides always sit opposite two equal angles, so the corners at M and D in this triangle match.

#7 Identify Subproblems 4.MD.C.6

The base angle at M is given as 40 deg, so the base angle at D inside the triangle is also 40 deg.

\angle ADM = \angle AMD = 40^\circ

Once one base angle of an isosceles triangle is known, the other base angle is the same number.

#7 Identify Subproblems 4.MD.C.7

The three angles of triangle AMD add to 180 deg, so the apex angle a at A is 180 deg minus the two base angles.

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

After taking away the two equal base angles, the leftover of 180 deg is the top angle at A.

Answer: 100 degrees

Review

With thin 40 deg base angles, the apex at A should be wide, and 100 deg (obtuse) fits. Check: 40 + 40 + 100 = 180 deg. The extra 20 deg at D (angle MDC) is consistent: angle ADC at the parallelogram corner is 40 + 20 = 60 deg, a valid parallelogram angle.

Use the parallelogram relations (tool 7): angle ADC = angle ADM + angle MDC = 40 + 20 = 60 deg, so angle DAB = 120 deg; combined with the isosceles triangle this cross-checks angle DAM = 100 deg.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing triangle AMD as isosceles from AM = AD.
4.MD.C.6 Measure angles in whole-number degrees using a protractor — Transferring the 40 deg base angle to the equal base angle at D.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180 deg triangle total to find the apex angle a.

💡 Equal sides mean equal base angles, so two 40 deg corners leave a 100 deg angle at A: just the triangle's leftover of 180 deg!