We are given a quilt design in which triangle ABC is congruent to triangle ADE.
We will find the measure of angle BCA. Recall that two figures are congruent if and only if their corresponding sides and corresponding angles are congruent. Let's identify the congruent sides and the congruent angles of the given triangles by using the order of the vertex labels in the congruence statement.
Congruent Triangles [0.5em]
△ A B C ≅ △ A D E [0.5em]
⇕ [0.5em]
cc
Congruent Sides [0.5em]
A B≅A D
B C≅D E
A C≅A E
and
cc
Congruent Angles [0.5em]
∠ B A C≅∠ D A E
∠ A B C≅∠ A D E
∠ B C A≅∠ D E A
Since ∠ BCA is congruent to ∠ DEA, the measure of ∠ BCA is equal to the measure of ∠ DEA. We are given that m∠ DEA is 59^(∘), so the measure of ∠ BCA is also 59^(∘).
Extra
Finding the Side Lengths of △ ADE
We can also use the diagram to find the length of each side of △ ADE. We know that corresponding sides of congruent figures are congruent. Let's use the information that the corresponding sides of △ ABC and △ ADE is congruent to find the side length of △ ADE.
cc
Congruent Sides [0.5em]
AB≅AD
BC≅DE
AC≅AE
Recall that congruent segments have the same length. We know the lengths of AB and BC, so we can find the lengths of DE and AD.
cc
AB=5 cm
BC=3 cm
⇒
cc
AD=5 cm
DE=3 cm
We can use the Pythagorean Theorem To find the length of AE because we are considering a right triangle. Note that DE and AD are the legs of the right triangle.
