5.1 Congruent Figures 

Congruent Figures:

Figures that have the Same Shape and the Same Size are considered to be Congruent.  In other words, if 2-polygons have the Same Size Sides AND the Same Size Angles in the Same Exact Order going around the polygon ( either clockwise or counter-clockwise ), then the 2-polygons are Congruent.

Corresponding sides and corresponding angles of polygons are those that are in the same position in two different polygons with the same number of sides. These corresponding parts are indicated by the names of the polygons. When naming congruent polygons, it is important that the order of the points, or vertices, in the names correspond.

If we were to look specifically at polygons, we would note that any polygon could be made up of Triangles, ( remember, that fact is used to calculate the measure of interior angle of a polygon ).  Here we can use that fact to help us determine if 2-polygons are congruent, so to simplify matters we are going to focus on how to determine if 2-triangles are congruent.

CPCTC:  “Corresponding Parts of Congruent Triangles are Congruent.”

Given that ΔXYZ and ΔKLM.are congruent, then the names of the triangles show that X corresponds to K, Y corresponds to L, and Z corresponds to M. Since we know which angles correspond with one another, then the sides enclosed by those angles must also correspond, thus side XY corresponds to side KL, side XZ corresponds to side KM, and side YZ corresponds to side LM.

Angle-Side-Angle (ASA) Triangle Congruence Theorem:

In any two triangles, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Example A

Use ASA congruence to determine the measure of the sides of ΔDEF.		Watch the video:
Solution:

Click here for a Quick Question:

Angle-Angle-Side (AAS) Triangle Congruence Theorem:

In any two triangles, if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Example B

Use AAS congruence to determine the missing measures of the given triangles if ΔABC ΔWXY and  mW = 120°, mB = 40°, side CA = 8, side XY = 12, side BA = 7.	Watch the video:
Solution:

Click here for a Quick Question:

Side-Side-Side (SSS) Triangle Congruence Theorem:

In any two triangles, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Example C

Can a Kite be broken up into 2-congruent triangles?		Watch the video:
Solution:

Click here for a Quick Question:

Side-Angle-Side (SAS) Triangle Congruence Theorem:

In any two triangles, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent by side-angle-side congruence.

Example D

Given the figure on the right, is ΔABC ΔDCB by SAS?		Watch the video:
Solution:

Click here for a Quick Question:

What about a Side-Side-Angle (SSA) Triangle Congruence Theorem?

Hypotenuse-Leg (HL) Right Triangle Congruence Theorem:

In any two right triangles, if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. The following are Corollaries of HL Right Triangle Congruence Theorem:      -  If a leg and an acute angle of one right triangle are congruent to a leg and        an acute angle of another right triangle, then the triangles are congruent.     -  If the hypotenuse and an acute angle of one right triangle are congruent tothe       hypotenuse and an acute angle of another right triangle, then the triangles are congruent.      -  If the two legs of one right triangle are congruent to the two legs of another       right triangle, then the triangles are congruent.

Example E

Given the rafter on the right, identify all triangles that can be congruent by HL congruence.		Watch the video:
Solution:

Click here for a Quick Question:

Practice Problems

If the given triangles are congruent, justify with a Triangle Congruence Theorem and indicate Corresponding Vertices.

Otherwise write, "Not Enough Information Given"

A.	B.
C.	D.
E.	F.

Assignment:

Remember to paste your results into your Jamboard for this assignment

1.  Use the Quick Question: Are triangles congruent? True/False      To do 5 of the True/False Triangle Congruence Questions.
2.  Use the Quick Question: Are triangles congruent? True/False      To do 5 of the True/False Triangle Congruence Questions.
3.  Use the Quick Question: Are triangles congruent? True/False      To do 5 of the True/False Triangle Congruence Questions.
4.  Use the Quick Question:  Are triangles congruent? True/False      To do 5 of the True/False Triangle Congruence Questions.
5.  Use the Quick Question: Are triangles congruent? True/False      To do 5 of the True/False Triangle Congruence Questions.
6  Use the Quick Question: Identify Congruence Theorem     To do 10 of the Identify Congruence Theorem
7.  In the figures on the right, AB FD & B D.      Identify the additional information need to justify that the triangles are congruent by: AAS Congruence Theorem ASA Congruence Theorem SAS Congruence Theorem SSS Congruence Theorem
8  Given the Kite on the right, use your knowledge of Polygons to explain     why ΔAEB ΔADC by either:  AAS Congruence Theorem ASA Congruence Theorem SAS Congruence Theorem SSS Congruence Theorem
9. Given the Regular Pentagon on the right, use your knowledge of Polygons to explain      why ΔAEV ΔEVI by either:  AAS Congruence Theorem ASA Congruence Theorem SAS Congruence Theorem SSS Congruence Theorem
10.  Given the Isosceles Trapezoid on the right, use your knowledge of Polygons and        Parallel Lines with Transversals to explain why ΔADU ΔUQA by either:  AAS Congruence Theorem ASA Congruence Theorem SAS Congruence Theorem SSS Congruence Theorem