Triangle angles sum to 180 degrees

4.MD.C.7 · take · grade 4

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

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $110^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $60^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 110 degrees. At A, the right-triangle angle CAD is 60 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 110 degrees (left triangle, at C).
Angle CAD = 60 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 110-degree angle, so 180 - 110. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 110 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 110^\circ = 70^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 60 degrees, ACD = 70 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

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

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 50 degrees

Review

Triangle ACD: 60 + 70 + 50 = 180 degrees, a valid triangle. The straight base at C: 110 + 70 = 180 degrees. Both checks hold, so a = 50 degrees.

Use the exterior-angle idea (tool 5/pattern): the 110-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 110 = 60 + a gives a = 50 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!