Author: Tim Heppner
Last saved by: MHE
Company: McGraw-Hill

Geometry	Chapter 4

Lesson 4-5

Example 1  Use ASA in Proofs
Write a paragraph proof.	
Given:	 bisects ∠BAC and ∠BDC.
Prove:	△BAC ≅ △BDC

Proof:
Since  bisects ∠BAC and ∠BDC, ∠BAD ≅ ∠CAD and ∠BDA ≅ ∠CDA.  ≅  by the Reflexive Property. By ASA, △BAC ≅ △BDC.

Example 2  Use AAS in Proofs
Write a flow proof.	
Given:	∠PSU ≅ ∠PTR
 ≅ 
Prove:	 ≅ 

Proof:

Example 3  Determine if Triangles Are Congruent
ENGINEERING  A roof support truss is shown at 
the right. Suppose the roof supports,  and , 
each measure 28 feet, BD = 16 feet, 
and m∠A and m∠C are 30. Determine whether 
△ABD ≅ △CBD. Justify your answer.

Explore	We are given three measurements of each 
triangle. We need to determine whether the 
two triangles are congruent.

Plan	Since m∠A = m∠C, ∠A ≅ ∠C. Likewise, AB = BC so  ≅  and BD = BD so  ≅ . Check each possibility using the five methods you know.

Solve	We are given information about side-side-angle (SSA). This is not a method to prove two triangles are congruent. 

Examine	Use a compass, protractor and ruler to draw	
a triangle with the given measurements.
For simplicity of measurements, we will 
use centimeters instead of feet, so the 
measurements will be proportional. 
•	Draw a segment 2.8 centimeters long.
•	At one end, draw an angle 30°. Extend 
the line longer than 2.8 centimeters.
•	At the other end of the segment, draw 
an arc with a radius of 1.6 centimeters 
such that it intersects the line.

	Notice that there are two possible segments that could determine the triangle. Since the given measurements do not lead to a unique triangle, we cannot show that the triangles are congruent.