Triangle Congruence
Recall that for triangles to be congruent all corresponding parts must be congruent, meaning all corresponding angles and all corresponding sides are congruent. That's six congruence statements! Can we prove triangles are congruent with fewer congruence statements? In this lesson we will look at ways to do this. Let’s see how this is done.
Recall that for triangles to be congruent all corresponding parts must be congruent, meaning all corresponding angles and all corresponding sides are congruent. That's six congruence statements! Can we prove triangles are congruent with fewer congruence statements? In this lesson we will look at ways to do this. Let’s see how this is done.
SSS: Side-Side-Side Congruence Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
SAS: Side-Angle-Side Congruence Postulate
If two sides and the included angle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
If two sides and the included angle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Example 1:
Decide whether the congruence statement is true. Explain your reasoning and identify the isometry.
Decide whether the congruence statement is true. Explain your reasoning and identify the isometry.
Solution:
a) True by SSS; reflection
b) True by SAS; rotation
c) Not True - not enough information
d) True by SSS; reflection
e) Not True - not enough information
f) True by SAS; reflection
Example 2:
a) True by SSS; reflection
b) True by SAS; rotation
c) Not True - not enough information
d) True by SSS; reflection
e) Not True - not enough information
f) True by SAS; reflection
Example 2:
And now a flow proof.
ASA: Angle-Side-Angle Congruence Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
AAS: Angle-Angle-Side Congruence Theorem
If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent.
If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent.
HL: Hypotenuse-Leg Theorem
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.
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.
For Examples 3-6
You are given congruent angles and sides shown by the tick and angle marks. Name the additional congruent sides or angles needed to prove that the triangles are congruent by each specific method.
Example 3:
You are given congruent angles and sides shown by the tick and angle marks. Name the additional congruent sides or angles needed to prove that the triangles are congruent by each specific method.
Example 3:
Solution:
a) \(\angle M\cong\angle P\) creates included side
b) \(\angle B\cong\angle O\) creates non-included side
Example 4:
a) \(\angle M\cong\angle P\) creates included side
b) \(\angle B\cong\angle O\) creates non-included side
Example 4:
Solution:
a) \(\angle PRS\cong\angle TIN\) creates non-included side
b) \(\angle P\cong\angle N\) creates included side
Example 5:
Prove: \(\triangle SAR\cong\triangle KAH\)
a) \(\angle PRS\cong\angle TIN\) creates non-included side
b) \(\angle P\cong\angle N\) creates included side
Example 5:
Prove: \(\triangle SAR\cong\triangle KAH\)
Solution:
a) \(\angle SAR\cong\angle KAH\)
b) \(\overline{SR}\cong\overline{HK}\)
Example 6:
Prove: \(\triangle MQP\cong\triangle ORN\)
a) \(\angle SAR\cong\angle KAH\)
b) \(\overline{SR}\cong\overline{HK}\)
Example 6:
Prove: \(\triangle MQP\cong\triangle ORN\)
Solution:
a) \(\angle PMQ\cong\angle NOR\)
b) \(\overline{PQ}\cong\overline{NR}\)
Example 7:
a) \(\angle PMQ\cong\angle NOR\)
b) \(\overline{PQ}\cong\overline{NR}\)
Example 7:
Solution:
This is a reflection. It helps to redraw the overlapping triangles as two separate triangles. Then you can see the congruent parts easier.
This is a reflection. It helps to redraw the overlapping triangles as two separate triangles. Then you can see the congruent parts easier.
Let's try a paragraph proof.
We are given \(\angle A\cong\angle I\) and \(\angle ARH\cong\angle IHR\). We also know that \(\overline{HR}\cong\overline{HR}\) by the Reflexive Property. Therefore the triangles are congruent by AAS.
Now let's try a flow proof.
We are given \(\angle A\cong\angle I\) and \(\angle ARH\cong\angle IHR\). We also know that \(\overline{HR}\cong\overline{HR}\) by the Reflexive Property. Therefore the triangles are congruent by AAS.
Now let's try a flow proof.