Download 2.5 Proving Angles Congruent

Warm-up 2.5
Fill in the blank
1. Perpendicular lines are two lines
that intersect to form _______.
2. An angle is formed by two rays with
the same endpoint. The endpoint is
called the _________ of the angle.
2.5 Proving Angles Congruent
Gonzales
Geometry
Definitions
• Vertical Angles
– Two angles whose sides form two pairs of
opposite rays
definitions
• Adjacent angles
– Two coplanar angles with a common side, a
common vertex, and no common interior
points
Definitions
• Complementary Angles
– Two angles whose measures have a sum of
90 degrees.
– Each angle is called the complement of the
other
Definitions
• Supplementary Angles
– Two angles whose measures have a sum of
180 degrees
– Each angle is called the supplement of the
other.
definitions
• Theorem
– Statements that have been mathematically
proven true.
• Paragraph proof
– Written as sentences in a paragraph
Theorems
• Vertical Angle Theorem
– Vertical angles are congruent
• Congruent Supplements theorem
– If two angles are supplements of the same angle (or
of congruent angles), then the two angles are
congruent.
• Congruent Complements theorem
– If two angles are complements of the same angle (or
of congruent angles), then the two angles are
congruent.
Theorems
• Theorem 2-4
– All right angles are congruent
• Theorem 2-5
– If two angles are congruent and
supplementary, then each is a right angle.
Proving Angles Congruent
LESSON 2-5
Additional Examples
Find the value of x.
The angles with labeled measures are vertical angles because their
sides are opposite rays. Apply the Vertical Angles Theorem to find x.
4x – 101 = 2x + 3
Vertical Angles Theorem
4x = 2x + 104
Addition Property of Equality
2x = 104
Subtraction Property of Equality
x = 52
Division Property of Equality
Quick Check
Proving Angles Congruent
LESSON 2-5
Additional Examples
Write a paragraph proof using the given, what you
are to prove, and the diagram.
Given: WX = YZ
Prove: WY = XZ
Proof: It is given that WX = YZ. Adding the middle segment, XY, to each
smaller segment gives WX + XY = YZ + XY.
By the Segment Addition Postulate, WX + XY = WY, and XY + YZ = XZ.
Using substitution, you get WY = XZ.
Quick Check
Examples
Example
Assignment
• Pg. 100
• 1-22
47-52