Chapter 8: Chapter 8

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 $\angle \mathrm{ADB} \cong \angle \mathrm{CBD}$.

Short Answer

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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.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

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, $\angle ADB \cong \angle CBD$.

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Chapter 8: Chapter 8

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Step 1: Identify the known side lengths and angles

Step 2: Use the SSS congruence postulate

Step 1: Identify the congruent triangles

Step 2: Apply congruent triangle properties

Step 1: Identify the known side lengths and angles

Step 2: Use the SAS congruence postulate

Step 1: Identify the congruent triangles

Step 2: Apply congruent triangle properties

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