Congruence Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Triangle PQR: ∠P=50°\angle P = 50°, ∠Q=60°\angle Q = 60°, PQ = 8 cm. Triangle XYZ: ∠X=50°\angle X = 50°, ∠Y=60°\angle Y = 60°, XY = 8 cm. Are they congruent? Which postulate applies?

Solution

  1. 1
    Step 1: Identify what is given — two angles and the included side for each triangle.
  2. 2
    Step 2: ∠P=∠X=50°\angle P = \angle X = 50°, ∠Q=∠Y=60°\angle Q = \angle Y = 60°, PQ=XY=8PQ = XY = 8 cm.
  3. 3
    Step 3: This matches the ASA (Angle-Side-Angle) postulate: two angles and the included side are equal.
  4. 4
    Step 4: Therefore △PQR≅△XYZ\triangle PQR \cong \triangle XYZ by ASA.

Answer

△PQR≅△XYZ\triangle PQR \cong \triangle XYZ by ASA.
ASA congruence works because two angles fix the shape of the triangle completely (the third angle is determined), and the included side fixes the scale. This uniquely determines the triangle up to congruence.

About Congruence

Two geometric figures are congruent if they have exactly the same size and shape, so one can be placed on the other perfectly.

Learn more about Congruence →

More Congruence Examples

Example 1 easy

Triangle ABC has sides 3 cm, 4 cm, 5 cm. Triangle DEF has sides 3 cm, 4 cm, 5 cm. Are they congruent

Example 2 medium

Two rectangles: Rectangle 1 has dimensions 4 cm × 6 cm. Rectangle 2 has dimensions 6 cm × 4 cm. Are

Example 3 easy

Are a square with side 5 cm and a rhombus with side 5 cm necessarily congruent? Explain why or why n

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