Question

find the missing angle:
find the measure of ∠a:
if two angles of one triangle are
______ to two angles of
another triangle, then
the triangles are
similar.
explain why these
triangles are similar.

Explanation:

Step1: Solve first missing angle

First, find the internal angle at $D$ adjacent to the missing exterior angle: $180^\circ - 95^\circ = 85^\circ$.
Use the exterior angle theorem: the exterior angle equals the sum of the two non-adjacent internal angles.
$\text{Missing angle} = 85^\circ + 15^\circ = 100^\circ$

Step2: Solve for $x$ (angle 4A)

Sum of triangle angles is $180^\circ$.
$5x + 5x + 130^\circ = 180^\circ$
$10x = 180^\circ - 130^\circ = 50^\circ$
$x = \frac{50^\circ}{10} = 5^\circ$
Measure of angle 4A: $5x = 5\times5^\circ = 25^\circ$

Step3: Fill similarity blank

Recall AA similarity criterion: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Step4: Explain triangle similarity

First, vertical angles at the intersection are congruent. Triangles have a pair of congruent $100^\circ$ angles, plus congruent vertical angles. By AA (Angle-Angle) similarity, the triangles are similar.

Answer:

1. Missing angle: $100^\circ$
2. Measure of angle 4A: $25^\circ$
3. Blank: congruent
4. The triangles share a pair of congruent vertical angles at their intersection, and both have a $100^\circ$ angle. By the Angle-Angle (AA) similarity criterion, the two triangles are similar.