G_Congruence_8_Lesson 7

Line and Angles Overview

Browse the course units below.

Congruence

Introduction

SSS

Introduction

ASA

Introduction

SAS

Introduction

AAS

Intro

RHS

Introduction

Test 1

MCQ

Test 2

Drag and Drop

Test 3

Axioms MCQ

Test 4

BLE 9(b)

Test 4

Mooc

Congruence

In geometry, two objects are congruent if one of them is placed on the other, and they exactly coincide. Congruent objects are equal in all respects. They have same shape, same size, and same area.

For example, two figures drawn on a piece of paper are congruent if they can be cut out and matched up perfectly (where flipping the paper over is allowed). The symbol used for congruence is \(\cong\).

For example, the sentence \(\triangle ABC \cong \triangle PQR\) is read as ‘triangle \(\triangle ABC\) is congruent to triangle \(\triangle PQR\).

The below shows the actual measurements of both triangles \(\triangle ABC\) and \(\triangle PQR\)

Corresponding Sides	Corresponding Angles
AB = PQ	∠A = ∠P
BC = QR	∠B = ∠Q
AC = PR	∠C = ∠R

Congruence Axiom

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Two traingles are congruent if following axioms are hold

1. SSS
2. ASA
3. SAS
4. AAS
5. RHS

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SSS

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Given		Results
Sides	Angles	Sides	Angles
AB=PQ			∠A = ∠P
BC = QR			∠B = ∠Q
AC = PR			∠C = ∠R

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ASA

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Given		Results
Sides	Angles	Sides	Angles
AB=PQ	∠A = ∠P
	∠B = ∠Q	BC = QR
		AC = PR	∠C = ∠R

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SAS

If two sides and the included angle of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Given		Results
Sides	Angles	Sides	Angles
AB=PQ	∠A = ∠P
AC=PR		BC = QR	∠B = ∠Q
			∠C = ∠R

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AAS

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Given		Results
Sides	Angles	Sides	Angles
BC=QR	∠A = ∠P
	∠B = ∠Q	AB=BC
		AC=PR	∠C = ∠R

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RHS

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Given		Results
Sides	Angles	Sides	Angles
BC=QR	∠A = ∠P
AB=PQ			∠B = ∠Q
		AC=PR	∠C = ∠R

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Test your Understanding: Quiz 1

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Test your Understanding: Quiz 2

Drag and drop each term into the correct category.

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Draggable Terms

Congruence

NOT Congruence

Test your Understanding: Quiz 3

Based on the information provided, what congruence theorem proves that △ABC ≅ △PQR?

1. In the given figure, if \(\triangle ABC \cong \triangle PQR\), find the values of \(x\) and \(y\). [2A]
2. Prove that \(\triangle AOC \cong \triangle BOD\) from the given figure. [2A]
3. By which axiom is \(\triangle ABO \cong \triangle DCO\). IF \(AO=10\), find the value of \(DO\) [2A]
4. Write any two facts with figure, so that two triangles are congruent. [2A]
5. By which axiom \(\triangle ABC \cong \triangle DEF\) are congruent, write the corresponding side of AB. [2A]
6. By which axiom \(\triangle ABC \cong \triangle DEF\) are congruent, write the corresponding side of AB. [2A]
7. In the given figure, Y is the midpoint of XZ, by which axiom are \(\triangle WXY \cong \triangle WYZ\)? If WX = 5 cm, what is the length of WZ?[2HA]
8. In parallelogram ABCD, BD is the B diagonal. By which axiom are \(\triangle ABD \cong \triangle CDB\)?[2HA]
9. Write the equal angles in the figure alongside if \(\triangle ABD \cong \triangle ACD\).[2HA]
10. In the adjoining figure, \(\triangle ABC \cong \triangle DEF\), find the value of a. [2HA]
11. Show that \(\triangle ADB \cong \triangle ADC\).[2HA]
12. In the given figure PQ = PR and QM = NR, prove that \(\triangle PMQ \cong \triangle PNR \). [2HA]
13. If BC = DE, by which axiom are \(\triangle ABC \cong \triangle ADE \)?[2HA]
14. In the given figure, if XY = XZ and YA = AZ, show that \(\triangle XAY \cong \triangle XAZ \). [2HA]
15. In the adjoining figure, if \( \angle ABD = \angle BAC\) and BD = AC, then prove that: \(\triangle ABC \cong \triangle ABD\). [2HA]
16. Identify the corresponding sides of congruent triangles given below. [2HA]