Key Concepts
- Identify congruent parts
Congruent Triangles
Introduction
What are congruent figures?
Two figures are said to be congruent if they have the same corresponding side lengths and the corresponding angles.
![What are congruent figures?](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1422.png)
What are similar figures?
Any two figures have the same shape, but their size is not the same.
![What are similar figures?](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1423.png)
Identify congruent parts
Congruence Statement:
![Congruence Statement:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1442.png)
From the above figures, the corresponding sides and the corresponding angles of both the triangles are equal.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1424.png)
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1425.png)
Properties of congruence:
The following properties define an equivalence relation.
- Reflexive Property – For all angles of A, ∠A≅∠A. An angle is congruent to itself.
- Symmetric Property – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.
- Transitive Property – For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent.
Example 1: Identify the pairs of congruent corresponding parts in the figure.
![Example 1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1426.png)
Solution: From the given figure,
∆JKL ≌ ∆TSR
Corresponding angles: ∠J ≌ ∠T, ∠K ≌ ∠S, ∠L ≌ ∠R
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1428.png)
Example 2: Find the values of x and y in the diagram using properties of congruent figures if DEFG ≌ SPQR.
![Example 2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1427-1024x293.png)
Solution:
Given that DEFG ≌ SPQR,
We know that FG ≌ QR.
⇒ FG = QR
12=2x−4
16=2x
8=x
Since ∠ F ≌ ∠Q.
m∠F= m∠Q
68°=(6y+x)°
68=6y+8
6y=68−8
y=606=10
Third angles theorem:
If two angles of one triangle are congruent to two angles of another triangle, then the third angle is also congruent.
![Third angles theorem:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1429-1024x312.png)
Given:
∠ A ≅ ∠D,
∠B ≅ ∠E.
To Prove:
∠ C ≅ ∠F.
Proof:
If ∠ A ≅ ∠D, and ∠B ≅ ∠E (given)
m∠ A = m∠D, m∠ B = m∠E (Congruent angles)
m∠A + m∠B + m∠C = 180° (Triangle sum theorem)
m∠D + m∠E + m∠F = 180°
m∠A + m∠B + m∠C = m∠D + m∠E + m∠F (Substitution property)
m∠D + m∠E + m∠C = m∠D + m∠E + m∠F (Substitution property)
m∠C = m∠F (Subtraction property of equality)
∠C ≅ ∠F. (Congruent angles)
Example 3: Find
m∠BDC
in the given figure.
![Example 3](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1430.png)
Solution:
From the given figure, we have
∠A ≅ ∠B and ∠ADC ≅∠BCD
⇒∠ACD ≅ ∠BDC (Third angles theorem)
m∠ACD + m∠CAN + m∠CDN = 180° (Triangle sum theorem)
m∠ACD = 180° – 45° – 30° = 105°.
m∠ACD = m∠BDC = 105°. (Congruent angles)
Properties of congruent triangles:
- Reflexive property of congruent triangles:
![Properties of congruent triangles:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1432.png)
For any triangle ABC, ∆ABC ≌ ∆ABC.
- Symmetric property of congruent triangles:
![Symmetric property of congruent triangles:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1431-1024x637.png)
If ∆ABC ≌ ∆DEF, then ∆DEF ≌ ∆ABC.
- Transitive property of congruent triangles:
![Transitive property of congruent triangles:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1432.png)
If ∆ABC ≌ ∆DEF and ∆DEF ≌ ∆JKL, then ∆ABC ≌ ∆JKL.
Example 4: Prove that the triangles are congruent.
Solution:
Given: AD ≌ CB, DC ≌ BA, ∠ ACD ≅ ∠ CAB, ∠ CAD ≅ ∠ ACB
![Example 4](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1433.png)
To prove: ∆ACD ≅ ∆CAB
proof:
a. Use the reflexive property to show that AC ≌ AC
b. Use the third angles theorem to show that ∠ B ≅ ∠ D
Plan in action:
![Plan in action:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1434-1024x498.png)
Exercise
- Identify all pairs of congruent corresponding parts in the following figure:
![Identify all pairs of congruent corresponding parts in the following figure:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1450.png)
- Find the value of x in the given figure.
![Find the value of x in the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1451.png)
- Find the values of x and y in the diagram.
![Find the values of x and y in the diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1452.png)
- Write the proof with the help of a diagram.
![Write the proof with the help of a diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1454.png)
Given: WX ⊥ VZ at Y, Y is the midpoint of WX, VW ≅ VX, and VZ bisects ∠WVX.
Proof: ∆VWY ≅ ∆VXY.
- Find from the given figure.
![Find from the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1455.png)
- Find the values of x and y in the diagram.
![Find the values of x and y in the diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1456.png)
- Find the value of x in the figure.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1457.png)
- Write the congruence statements for the given figure.
![Write the congruence statements for the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1458.png)
- Identify the congruent corresponding parts.
![Identify the congruent corresponding parts.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1459.png)
- Find in the figure.
![Find in the figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1460.png)
What have we learned
- Understand congruent of two figures.
- Understand the congruence statements.
- Find corresponding angles and corresponding sides.
- Identify congruent parts for the given triangles.
- Find the values using properties of congruent figures.
- Understand third angles theorem.
- Understand properties of congruent triangles.
- Solve problems on congruent figures.
Summary
Properties of congruence:
The following properties define an equivalence relation.
1.Reflexive property – For all angles of A, ∠A≅∠A. An angle is congruent to itself.
2. Symmetric property – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.
3.Transitive property – For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent.
![congruent-triangles](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2022%2F09%2Fcongruent-triangles.png&w=1920&q=50)
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