Explanation Given information: AB=BC, BP and CP are angle bisectors Formula used: Properties of triangles. Proof: A B = B C ( g i v e n ) ∴ ∠ A B C = ∠ A C B ( A n g l e s o p p o s i t e t o e q u a l s i d e s a r e e q u a l ) ∴ 1 2 ∠ A B C = 1 2 ∠ A C B . ∴ ∠ P B M = ∠ P C M ( B P & P C a r e a n g l e b i s e c t o r ) ∴ P B = P C ( S i d e s o p p o s i t e s t o e q u a l a n g l e s a r e e q u a l )

91st Edition

ISBN: 9780866099653

Author: Richard Rhoad, George Milauskas, Robert Whipple

Publisher: McDougal Littell

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Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 4.2, Problem 12PSB

To determine

To Prove: $ΔBPC$ is isosceles.

Expert Solution & Answer

Explanation of Solution

Given information: AB=BC, BP and CP are angle bisectors

Formula used: Properties of triangles.

Proof:

Geometry For Enjoyment And Challenge, Chapter 4.2, Problem 12PSB

$AB=BC (given)∴∠ABC=∠ACB (Angles opposite to equal sides are equal)∴12∠ABC=12∠ACB. ∴∠PBM=∠PCM (BP & PC are angle bisector)∴PB=PC (Sides opposites to equal angles are equal)$

Chapter 4.2, Problem 11PSB

Chapter 4.2, Problem 13PSC

Chapter 4 Solutions

Geometry For Enjoyment And Challenge

Ch. 4.1 - Prob. 1PSA Ch. 4.1 - Prob. 2PSA Ch. 4.1 - Prob. 3PSA Ch. 4.1 - Prob. 4PSA Ch. 4.1 - Prob. 5PSA Ch. 4.1 - Prob. 6PSA Ch. 4.1 - Prob. 7PSA Ch. 4.1 - Prob. 8PSA Ch. 4.1 - Prob. 9PSB Ch. 4.1 - Prob. 10PSB

Ch. 4.1 - Prob. 11PSB Ch. 4.1 - Prob. 12PSB Ch. 4.1 - Prob. 13PSB Ch. 4.1 - Prob. 14PSB Ch. 4.1 - Prob. 15PSB Ch. 4.1 - Prob. 16PSC Ch. 4.1 - Prob. 17PSC Ch. 4.1 - Prob. 18PSC Ch. 4.2 - Prob. 1PSA Ch. 4.2 - Prob. 2PSA Ch. 4.2 - Prob. 3PSA Ch. 4.2 - Prob. 4PSA Ch. 4.2 - Prob. 5PSA Ch. 4.2 - Prob. 6PSA Ch. 4.2 - Prob. 7PSA Ch. 4.2 - Prob. 8PSB Ch. 4.2 - Prob. 9PSB Ch. 4.2 - Prob. 10PSB Ch. 4.2 - Prob. 11PSB Ch. 4.2 - Prob. 12PSB Ch. 4.2 - Prob. 13PSC Ch. 4.2 - Prob. 14PSC Ch. 4.2 - Prob. 15PSC Ch. 4.3 - Prob. 1PSA Ch. 4.3 - Prob. 2PSA Ch. 4.3 - Prob. 3PSA Ch. 4.3 - Prob. 4PSA Ch. 4.3 - Prob. 5PSA Ch. 4.3 - Prob. 6PSA Ch. 4.3 - Prob. 7PSA Ch. 4.3 - Prob. 8PSB Ch. 4.3 - Prob. 9PSB Ch. 4.3 - Prob. 10PSB Ch. 4.3 - Prob. 11PSB Ch. 4.3 - Prob. 12PSB Ch. 4.3 - Prob. 13PSB Ch. 4.3 - Prob. 14PSC Ch. 4.3 - Prob. 15PSC Ch. 4.3 - Prob. 16PSC Ch. 4.4 - Prob. 1PSA Ch. 4.4 - Prob. 2PSA Ch. 4.4 - Prob. 3PSA Ch. 4.4 - Prob. 4PSA Ch. 4.4 - Prob. 5PSA Ch. 4.4 - Prob. 6PSA Ch. 4.4 - Prob. 7PSA Ch. 4.4 - Prob. 8PSA Ch. 4.4 - Prob. 9PSA Ch. 4.4 - Prob. 10PSA Ch. 4.4 - Prob. 11PSB Ch. 4.4 - Prob. 12PSB Ch. 4.4 - Prob. 13PSB Ch. 4.4 - Prob. 14PSB Ch. 4.4 - Prob. 15PSB Ch. 4.4 - Prob. 16PSB Ch. 4.4 - Prob. 17PSB Ch. 4.4 - Prob. 18PSB Ch. 4.4 - Prob. 19PSB Ch. 4.4 - Prob. 20PSC Ch. 4.4 - Prob. 21PSC Ch. 4.4 - Prob. 22PSC Ch. 4.4 - Prob. 23PSC Ch. 4.4 - Prob. 24PSC Ch. 4.5 - Prob. 1PS Ch. 4.5 - Prob. 2PS Ch. 4.5 - Prob. 3PS Ch. 4.5 - Prob. 4PS Ch. 4.5 - Prob. 5PS Ch. 4.6 - Prob. 1PSA Ch. 4.6 - Prob. 2PSA Ch. 4.6 - Prob. 3PSA Ch. 4.6 - Prob. 4PSA Ch. 4.6 - Prob. 5PSA Ch. 4.6 - Prob. 6PSA Ch. 4.6 - Prob. 7PSA Ch. 4.6 - Prob. 8PSA Ch. 4.6 - Prob. 9PSA Ch. 4.6 - Prob. 10PSA Ch. 4.6 - Prob. 11PSA Ch. 4.6 - Prob. 12PSB Ch. 4.6 - Prob. 13PSB Ch. 4.6 - Prob. 14PSB Ch. 4.6 - Prob. 15PSB Ch. 4.6 - Prob. 16PSB Ch. 4.6 - Prob. 17PSC Ch. 4.6 - Prob. 18PSC Ch. 4 - Prob. 1RP Ch. 4 - Prob. 2RP Ch. 4 - Prob. 3RP Ch. 4 - Prob. 4RP Ch. 4 - Prob. 5RP Ch. 4 - Prob. 6RP Ch. 4 - Prob. 7RP Ch. 4 - Prob. 8RP Ch. 4 - Prob. 9RP Ch. 4 - Prob. 10RP Ch. 4 - Prob. 11RP Ch. 4 - Prob. 12RP Ch. 4 - Prob. 13RP Ch. 4 - Prob. 14RP Ch. 4 - Prob. 15RP Ch. 4 - Prob. 16RP Ch. 4 - Prob. 17RP Ch. 4 - Prob. 18RP Ch. 4 - Prob. 19RP Ch. 4 - Prob. 20RP Ch. 4 - Prob. 21RP Ch. 4 - Prob. 22RP

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To Prove: Δ B P C is isosceles.