### Geometry Proof Based on Circle and Inscribed Triangles**Problem Statement:**Given that $ AB \cong CD $, and $\triangle BCA$ and $\triangle CBD$ are inscribed, prove the following:- $\triangle BCA \sim \triangle CBD$- $\angle ABC \cong \angle DCB$**Diagram Description:**The diagram consists of a circle with four points $A$, $B$, $C$, and $D$ on its circumference. The points are connected by chords such that:- $AB$ and $CD$ intersect each other inside the circle.- Points $A$ and $C$ are endpoints of one chord.- Points $B$ and $D$ are endpoints of another chord.The triangles $\triangle BCA$ and $\triangle CBD$ are formed by these intersections.**Directions:**1. Identify the given congruent segments $ AB \cong CD $.2. Demonstrate the similarity of the triangles $\triangle BCA$ and $\triangle CBD$ by comparing their corresponding angles.3. Show that $\angle ABC \cong \angle DCB$.This exercise involves understanding and demonstrating the properties of inscribed triangles within a circle, as well as proving the relationships between the angles and sides of these triangles based on the given conditions.

To prove - angle ABC = angle DCB Triangle BCA= triangle CBD

Answered: 4 Given: AB PC; A BCA and A CBD…

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4 Given: AB PC; A BCA and A CBD insubscribe in circle prove: A BCA GA ALABC LDCB 000 ACBD ん A

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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### Geometry Proof Based on Circle and Inscribed Triangles

**Problem Statement:**
Given that $ AB \cong CD $, and $\triangle BCA$ and $\triangle CBD$ are inscribed, prove the following:
- $\triangle BCA \sim \triangle CBD$
- $\angle ABC \cong \angle DCB$

**Diagram Description:**
The diagram consists of a circle with four points $A$, $B$, $C$, and $D$ on its circumference. The points are connected by chords such that:
- $AB$ and $CD$ intersect each other inside the circle.
- Points $A$ and $C$ are endpoints of one chord.
- Points $B$ and $D$ are endpoints of another chord.
The triangles $\triangle BCA$ and $\triangle CBD$ are formed by these intersections.

**Directions:**
1. Identify the given congruent segments $ AB \cong CD $.
2. Demonstrate the similarity of the triangles $\triangle BCA$ and $\triangle CBD$ by comparing their corresponding angles.
3. Show that $\angle ABC \cong \angle DCB$.

This exercise involves understanding and demonstrating the properties of inscribed triangles within a circle, as well as proving the relationships between the angles and sides of these triangles based on the given conditions.

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