Hewie thinks that \(\triangle MAT\cong\triangle THM\) by SSS. Dewey disagrees and thinks that \(\triangle MAT\cong\triangle MHT\) by SAS. Louie thinks that the triangles are congruent by HL. Who do you agree with and why?
All three ducks think that \(\angle AMT\cong\angle HTM\), and \(\overline{MA}\parallel\overline{HT}\) because the triangles are congruent. Are they correct?
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We now know that triangles are special. They only require three pairs of corresponding congruent parts to prove that the triangles are congruent (SSS, SAS, ASA, AAS, HL). Because of that, we have yet another theorem that comes AFTER proving two triangles are congruent.
Let’s take a look at the following example:
Let’s take a look at the following example:
Solution:
We since all pairs of corresponding sides are congruent, we can prove the triangles to be congruent by SSS! Now think back to Example 1. Since we have used the three pairs of sides congruent to prove the triangles are congruent, what other corresponding parts of the triangle are congruent? |
The other parts that must be congruent are each of the corresponding angles. That means: \(\angle A\cong\angle L\), \(\angle M\cong\angle C\), and \(\angle J\cong\angle G\) . Notice we didn’t originally use the angles to prove that the triangles are congruent. We used other parts of the triangles to prove the triangles congruent, and then all the other corresponding must be congruent too! This brings us to our knew theorem:
Theorem
If triangles are congruent, then all other corresponding segments or angles are congruent. We can also say “If triangles are congruent, then corresponding parts are congruent” (CPCTC). Notice, however, that the hypothesis of this theorem begins with “triangles are congruent”. So we can only use this theorem if we are given or have proven triangles to be congruent.
If triangles are congruent, then all other corresponding segments or angles are congruent. We can also say “If triangles are congruent, then corresponding parts are congruent” (CPCTC). Notice, however, that the hypothesis of this theorem begins with “triangles are congruent”. So we can only use this theorem if we are given or have proven triangles to be congruent.
Example 2:
Use the tick marks in the diagram below to: a) State the method used to prove the triangles congruent b) Now, what other parts are also congruent? Solution: a) A reflection will map one triangle onto the other. The tick marks in the diagram show that \(\overline{ET}\cong\overline{AT}\) and (\overline{TX}\cong\overline{TR}\). Also, \(\angle ETX\cong\angle ATR\) because vertical angles are congruent. Therefore, \(\triangle TEX\cong\triangle TAR\) by SAS. |
b) We will list out the other corresponding parts that are not marked on the diagram. Since we proved the triangles by SAS, that means there is one more pair of sides and two pairs of angles that will also be congruent: \(\overline{EX}\cong\overline{AR}\), \(\angle E\cong\angle A\), and \(\angle X\cong\angle R\). The reason why we can add these extra congruence statements is because we first proved the triangles congruent, so then all other corresponding parts are congruent (CPCTC).
Solution Method 1:
Let’s watch the video for the solution written in flow format!
Let’s watch the video for the solution written in flow format!
Solution Method 2:
Paragraph format
We know that \(\overline{BE}\parallel\overline{AN}\) because it’s given. Since the lines are parallel they form AIA\(\cong\) so \(\angle EBA\cong\angle NAB\). We also know that \(\angle E\cong\angle N\) because it is also given. From the diagram, we can see that \(\overline{BA}\cong\overline{AB}\) by the reflexive property. Since we have two pairs of angles and one pair of non-included sides congruent, we know that \(\triangle BNA\cong\triangle AEB\) by AAS. Since the triangles are congruent the corresponding sides are congruent, \(\overline{BN}\cong\overline{AE}\).
Solution Method 3:
Two column format
Paragraph format
We know that \(\overline{BE}\parallel\overline{AN}\) because it’s given. Since the lines are parallel they form AIA\(\cong\) so \(\angle EBA\cong\angle NAB\). We also know that \(\angle E\cong\angle N\) because it is also given. From the diagram, we can see that \(\overline{BA}\cong\overline{AB}\) by the reflexive property. Since we have two pairs of angles and one pair of non-included sides congruent, we know that \(\triangle BNA\cong\triangle AEB\) by AAS. Since the triangles are congruent the corresponding sides are congruent, \(\overline{BN}\cong\overline{AE}\).
Solution Method 3:
Two column format
Quick Check
1) Use the tick marks in the diagram below to:
a) State the method used to prove the triangles congruent
b) Now, what other parts are also congruent?
1) Use the tick marks in the diagram below to:
a) State the method used to prove the triangles congruent
b) Now, what other parts are also congruent?