Suggested languages for you:

Americas

Europe

Problem 120

# LET $$A B C D$$ be a quadrilateral in which $$A D=B C$$ and $$\mathrm{AB}=\mathrm{CD}$$. Let its diagonals, $$\underline{\mathrm{AC}}$$ and $$\underline{\mathrm{BD}}$$, intersect at point E. a) Prove that $\triangle \mathrm{ABC} \cong \Delta \mathrm{CDA} ; \mathrm{b}$$) Prove that$$\angle \mathrm{DAC} \cong \angle \mathrm{BCA}$; c) Prove that $$\triangle \mathrm{ABD} \cong \Delta \mathrm{CDB} ; \mathrm{d}$$ ) Prove that $$\angle \mathrm{ADB} \cong \angle \mathrm{CBD}$$.

Expert verified
In summary, we proved that: a) $$\triangle ABC \cong \triangle CDA$$ using the SSS congruence postulate. b) $$\angle DAC \cong \angle BCA$$ as corresponding angles of congruent triangles. c) $$\triangle ABD \cong \triangle CDB$$ using the SAS congruence postulate. d) $$\angle ADB \cong \angle CBD$$ as corresponding angles of congruent triangles.
See the step by step solution

## Step 1: Identify the known side lengths and angles

We know that $$AD = BC$$ and $$AB = CD$$.

## Step 2: Use the SSS congruence postulate

We see that $$AB = CD$$, $$AD = BC$$, and $$AC = AC$$ (shared side). By the SSS postulate, we can conclude that $$\triangle ABC \cong \triangle CDA$$. #b) Prove that angle DAC is congruent to angle BCA#

## Step 1: Identify the congruent triangles

From part a, we know that $$\triangle ABC \cong \triangle CDA$$.

## Step 2: Apply congruent triangle properties

Since $$\triangle ABC \cong \triangle CDA$$, their corresponding angles are also congruent. Therefore, $$\angle DAC \cong \angle BCA$$. #c) Prove that triangle ABD is congruent to triangle CDB#

## Step 1: Identify the known side lengths and angles

We know that $$AB = CD$$, and from the result in part b, we know that $$\angle DAC \cong \angle BCA$$.

## Step 2: Use the SAS congruence postulate

We know that $$AB = CD$$, $$\angle DAC \cong \angle BCA$$, and $$BD$$ is a common side for both triangles. By the Side-Angle-Side (SAS) congruence postulate, we can conclude that $$\triangle ABD \cong \triangle CDB$$. #d) Prove that angle ADB is congruent to angle CBD#

## Step 1: Identify the congruent triangles

From part c, we know that $$\triangle ABD \cong \triangle CDB$$.

## Step 2: Apply congruent triangle properties

Since $$\triangle ABD \cong \triangle CDB$$, their corresponding angles are also congruent. Therefore, $$\angle ADB \cong \angle CBD$$.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner