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Triangles - Congruence Theorems - Prove triangle properties and evaluate validity of converse statements

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Congruence of triangles refers to figures that have the same shape and size. Two triangles △ABC\triangle ABC and △PQR\triangle PQR are congruent (denoted △ABC≅△PQR\triangle ABC \cong \triangle PQR) if their corresponding sides and angles are equal.

Two identical triangles ABC and PQR illustrating congruence.
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The SAS (Side-Angle-Side) Congruence Rule states that two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

SAS diagram showing two sides and the angle between them.
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The ASA (Angle-Side-Angle) Rule and AAS (Angle-Angle-Side) Rule: ASA requires the side to be included between two angles, whereas AAS states that congruence holds even if the equal side is not between the two equal angles.

ASA diagram showing two angles and the side between them.
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Converse statements in geometry: The converse of a theorem is formed by swapping the hypothesis and the conclusion. For example, if AB=AC⇒∠B=∠CAB = AC \Rightarrow \angle B = \angle C, the converse is: if ∠B=∠C⇒AB=AC\angle B = \angle C \Rightarrow AB = AC. Not all converse statements are true, but for isosceles triangles and congruence criteria, they often hold.

Isosceles triangle showing equal sides and potential for converse properties.

📐Formulae

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ

Exterior ∠ACD=∠A+∠B\text{Exterior } \angle ACD = \angle A + \angle B

AB+BC>ACAB + BC > AC (Triangle Inequality)

In △ABC, if AB=AC⇒∠C=∠B\text{In } \triangle ABC, \text{ if } AB = AC \Rightarrow \angle C = \angle B

Area of a triangle=12×base×height\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}

Perimeter of △ABC=AB+BC+CA\text{Perimeter of } \triangle ABC = AB + BC + CA

💡Examples

Problem 1:

In △XYZ\triangle XYZ, the measure of ∠X=45∘\angle X = 45^\circ and ∠Y=75∘\angle Y = 75^\circ. Find the measure of ∠Z\angle Z.

Solution:

Step 1: Identify the Angle Sum Property, which states that ∠X+∠Y+∠Z=180∘\angle X + \angle Y + \angle Z = 180^\circ. Step 2: Substitute the known values: 45∘+75∘+∠Z=180∘45^\circ + 75^\circ + \angle Z = 180^\circ. Step 3: Simplify the equation: 120∘+∠Z=180∘120^\circ + \angle Z = 180^\circ. Step 4: Subtract 120∘120^\circ from both sides: ∠Z=180∘−120∘\angle Z = 180^\circ - 120^\circ. Step 5: Therefore, ∠Z=60∘\angle Z = 60^\circ.

Explanation:

This problem uses the fundamental property that all internal angles of any triangle must add up to exactly 180180 degrees.

Problem 2:

In an isosceles triangle ABCABC, AB=ACAB = AC and the vertex angle ∠A=50∘\angle A = 50^\circ. Find the measures of the base angles ∠B\angle B and ∠C\angle C.

Solution:

Step 1: Given AB=ACAB = AC, we know from the Isosceles Triangle Theorem that ∠B=∠C\angle B = \angle C. Let ∠B=∠C=x\angle B = \angle C = x. Step 2: Use the Angle Sum Property: ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ. Step 3: Substitute the values: 50∘+x+x=180∘50^\circ + x + x = 180^\circ. Step 4: Combine like terms: 50∘+2x=180∘50^\circ + 2x = 180^\circ. Step 5: Isolate xx: 2x=180∘−50∘⇒2x=130∘2x = 180^\circ - 50^\circ \Rightarrow 2x = 130^\circ. Step 6: Solve for xx: x=130∘2=65∘x = \frac{130^\circ}{2} = 65^\circ. Step 7: So, ∠B=65∘\angle B = 65^\circ and ∠C=65∘\angle C = 65^\circ.

Explanation:

The solution relies on two properties: first, that equal sides imply equal opposite angles in a triangle, and second, that all angles must sum to 180180 degrees.

Problem 3:

In △ABC\triangle ABC, ADAD is the perpendicular bisector of BCBC. Show that △ABC\triangle ABC is an isosceles triangle in which AB=ACAB = AC.

Triangle ABC with altitude AD bisecting base BC.

Solution:

In △ABD\triangle ABD and △ACD\triangle ACD:

  1. BD=CDBD = CD (Since ADAD bisects BCBC)
  2. ∠ADB=∠ADC=90∘\angle ADB = \angle ADC = 90^\circ (AD⊥BCAD \perp BC)
  3. AD=ADAD = AD (Common side)

Therefore, △ABD≅△ACD\triangle ABD \cong \triangle ACD by SAS rule. By CPCT (Corresponding Parts of Congruent Triangles), AB=ACAB = AC. Since two sides are equal, △ABC\triangle ABC is an isosceles triangle.

Explanation:

To prove a triangle is isosceles, we look for two congruent sub-triangles created by an altitude or median. Here, SAS is applicable because the shared side and the bisected base surround the 90∘90^\circ angle.

Problem 4:

Is the converse of the SSS congruence theorem true? Statement: 'If two triangles have equal areas, they are congruent.' Evaluate the validity.

Two different triangles with the same area showing they are not congruent.

Solution:

The statement 'If two triangles have equal areas, they are congruent' is FALSE.

Consider △1\triangle 1 with base 44 and height 33: Area=12×4×3=6\text{Area} = \frac{1}{2} \times 4 \times 3 = 6. Consider △2\triangle 2 with base 66 and height 22: Area=12×6×2=6\text{Area} = \frac{1}{2} \times 6 \times 2 = 6.

The areas are equal, but the side lengths (dimensions) are different, so the triangles are not congruent.

Explanation:

Congruence requires identical side lengths and angles. Area only requires the product of base and height to be equal. Multiple different triangle shapes can yield the same area value.