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In the diagram below, AABD ≈ ACBD. Solve for y. A 7 y B D y = 5 y = 35 y = 20 3y + 20 C

In the diagram below, AABD ≈ ACBD. Solve for y. A 7 y B D y = 5 y = 35 y = 20 3y + 20 C

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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In the diagram, triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent. The task is to solve for \( y \). The parallelogram is labeled with the following information: - Diagonal \( BD \) represents a common side of both triangles. - Side \( AB \)= \( 7y \) - Side \( DC = 3y + 20 \) Options for \( y \) are provided as: - \( y = 5 \) - \( y = 35 \) - \( y = 20 \) - \( y = 7 \) By setting \( AB = DC \), we solve the equation: \[ 7y = 3y + 20 \] Subtract \( 3y \) from each side: \[ 4y = 20 \] Divide by 4: \[ y = 5 \] Thus, the correct value of \( y \) is 5.
Transcribed Image Text:In the diagram, triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent. The task is to solve for \( y \). The parallelogram is labeled with the following information: - Diagonal \( BD \) represents a common side of both triangles. - Side \( AB \)= \( 7y \) - Side \( DC = 3y + 20 \) Options for \( y \) are provided as: - \( y = 5 \) - \( y = 35 \) - \( y = 20 \) - \( y = 7 \) By setting \( AB = DC \), we solve the equation: \[ 7y = 3y + 20 \] Subtract \( 3y \) from each side: \[ 4y = 20 \] Divide by 4: \[ y = 5 \] Thus, the correct value of \( y \) is 5.
### Geometry: Proving Triangle Congruence by SSS To determine which statement would prove the triangles \( \triangle ABC \) and \( \triangle RQP \) are congruent by the Side-Side-Side (SSS) postulate, let's first review the triangles depicted: #### Diagram Description: - **Triangle \( \triangle ABC \):** - Side \( AB \): A single line mark indicates congruence with another side. - Side \( AC \): Two line marks indicate congruence with another side. - **Triangle \( \triangle RQP \):** - Side \( PQ \): Two line marks indicate congruence with \( AC \). - Side \( RP \): A single line mark indicates congruence with \( AB \). To use the SSS postulate, we need to establish congruence between all three corresponding sides of the triangles. #### Statement Options: - \( \text{( ) } AB \cong PQ \) - \( \text{( ) } BC \cong PQ \) - \( \text{( ) } \angle P \cong \angle C \) - \( \text{( ) } \angle A \cong \angle R \) #### Correct Statement for Proving Congruence by SSS: Select the statement \( AB \cong RP \) that shows congruence between two sides with a single line mark in each triangle. To prove congruence by SSS, we need: - \( AB \cong RP \) (Already indicated by a single line mark) - \( AC \cong PQ \) (Indicated by two line marks) - Confirm the option \( BC \cong RQ \) since it's necessary for SSS. #### Conclusion: For \( \triangle ABC \cong \triangle RQP \) by SSS, confirm the statement \( \text{BC} \cong \text{RQ} \). ### Educational Note: When verifying triangle congruence using methods like SSS, always analyze congruence indicators on the triangles diagrammatically to validate the relationships between their corresponding sides.
Transcribed Image Text:### Geometry: Proving Triangle Congruence by SSS To determine which statement would prove the triangles \( \triangle ABC \) and \( \triangle RQP \) are congruent by the Side-Side-Side (SSS) postulate, let's first review the triangles depicted: #### Diagram Description: - **Triangle \( \triangle ABC \):** - Side \( AB \): A single line mark indicates congruence with another side. - Side \( AC \): Two line marks indicate congruence with another side. - **Triangle \( \triangle RQP \):** - Side \( PQ \): Two line marks indicate congruence with \( AC \). - Side \( RP \): A single line mark indicates congruence with \( AB \). To use the SSS postulate, we need to establish congruence between all three corresponding sides of the triangles. #### Statement Options: - \( \text{( ) } AB \cong PQ \) - \( \text{( ) } BC \cong PQ \) - \( \text{( ) } \angle P \cong \angle C \) - \( \text{( ) } \angle A \cong \angle R \) #### Correct Statement for Proving Congruence by SSS: Select the statement \( AB \cong RP \) that shows congruence between two sides with a single line mark in each triangle. To prove congruence by SSS, we need: - \( AB \cong RP \) (Already indicated by a single line mark) - \( AC \cong PQ \) (Indicated by two line marks) - Confirm the option \( BC \cong RQ \) since it's necessary for SSS. #### Conclusion: For \( \triangle ABC \cong \triangle RQP \) by SSS, confirm the statement \( \text{BC} \cong \text{RQ} \). ### Educational Note: When verifying triangle congruence using methods like SSS, always analyze congruence indicators on the triangles diagrammatically to validate the relationships between their corresponding sides.
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