We are asked to write a two-column proof of the statement that angles ∠ DAC and ∠ CBD are congruent. Before we do that, let's mark them in the diagram.
Quadrilateral ABCD is an isosceles trapezoid. This means that its legs and base angles are congruent. We can highlight these two facts in our diagram.
Let's summarize what we know about triangles △ DAC and △ CBD using a table.
Congruence
Justification
DA≅CB
Definition (legs of isosceles trapezoid ABCD)
∠ ADC ≅ ∠ BCD
Theorem 6.21 (base angles of isosceles trapezoid ABCD)
Now, segment DC is a common side of triangles △ DAC and △ CBD. By the Reflexive Property of Congruence, DC≅ DC. As a result, two sides and an included angle in △ DAC are congruent to two sides and an included angle in △ CBD.
cc
DA ≅ CB & Side
∠ ADC ≅ ∠ BCD & Included Angle
DC≅ DC & Side
According to the Side-Angle-Side (SAS) Congruence Postulate, this means that the triangles are congruent.
△ DAC≅ △ CBD
If two triangles are congruent, then their corresponding angles are also congruent. Angles ∠ DAC and ∠ CBD are corresponding angles, so they are congruent.
Completed Proof
We can summarize our findings in a two-column proof.
2
&Given:&& ABCD is an isosceles trapezoid with
& && AB∥DCandAD≅BC
&Prove:&& ∠ DAC≅∠ CBD
Proof:
Statements
Reasons
1.
ABCD is an isosceles trapezoid AB∥DC and AD≅BC
1.
Given
2.
∠ADC≅∠BCD
2.
Base angles of an isosceles trapezoid (Theorem 6.21)
