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Step-by-step answers for all textbook questions in this chapter

Samacheer Kalvi 7th Maths (Term 2) Geometry Exercise Solutions

How to Use This Page

This page provides complete, step-by-step solutions for every exercise in Chapter 4 — Geometry of the Samacheer Kalvi 7th Maths Term 2 textbook. Work through each problem on your own first, then check your working against the solutions below. Focus on understanding why each step is taken, not just the final answer.

Key Concepts (Quick Recap)

Angle sum property: The sum of all three interior angles of any triangle is always 180 degrees
Types of triangles by sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal)
Types of triangles by angles: Acute (all angles < 90 degrees), Right (one angle = 90 degrees), Obtuse (one angle > 90 degrees)
Isosceles triangle theorem: Angles opposite equal sides are equal (and vice versa)
Congruence criteria: SSS, SAS, ASA, and RHS (for right-angled triangles)
RHS congruence: Two right-angled triangles are congruent if their hypotenuse and one side are equal
CPCT: Corresponding Parts of Congruent Triangles are equal

Exercise 4.1 — Angle Sum Property of Triangles

Q1. "The sum of any two angles of a triangle is always greater than the third angle." Is this true?

Solution

No, this statement is not always true.

Counterexample: Consider an isosceles right triangle with angles 90 degrees, 45 degrees, 45 degrees.

Here, 45 degrees + 45 degrees = 90 degrees, which equals the third angle (not greater than it).

Therefore, the sum of two angles of a triangle need not always be greater than the third angle.

AnswerThe statement is false. A counterexample is a right isosceles triangle (90 degrees, 45 degrees, 45 degrees) where 45 degrees + 45 degrees = 90 degrees.

Q2. Three angles of a triangle are in ratio 1:2:1. Find the angles and classify the triangle.

Solution

Let the angles be x, 2x, and x.

By the angle sum property:

x + 2x + x = 180 degrees

4x = 180 degrees

x = 45 degrees

So the angles are: 45 degrees, 90 degrees, 45 degrees.

Classification:

Two angles are equal (45 degrees each), so it is an isosceles triangle
One angle is 90 degrees, so it is also a right-angled triangle

AnswerAngles are 45 degrees, 90 degrees, 45 degrees. It is a right-angled isosceles triangle.

Q3. Find the values of x and y from the given figures:

(i)

GivenOne exterior angle = 120 degrees, and two interior angles are x and 50 degrees.

Solution

The exterior angle (120 degrees) and the interior angle y are supplementary (they form a linear pair):

y + 120 degrees = 180 degrees

y = 180 degrees − 120 degrees = 60 degrees

Now, using the angle sum property in the triangle:

x + y + 50 degrees = 180 degrees

x + 60 degrees + 50 degrees = 180 degrees

x + 110 degrees = 180 degrees

x = 70 degrees

Answerx = 70 degrees, y = 60 degrees

(ii)

GivenThree angles in the triangle are 50 degrees, 60 degrees, and y. An exterior angle adjacent to y is x.

Solution

Using the angle sum property:

50 degrees + 60 degrees + y = 180 degrees

110 degrees + y = 180 degrees

y = 70 degrees

Since x and y form a linear pair:

x + y = 180 degrees

x + 70 degrees = 180 degrees

x = 110 degrees

Answery = 70 degrees, x = 110 degrees

Q4. Two angles of a triangle are 30 degrees and 80 degrees. Find the third angle.

Solution

By the angle sum property, the sum of all three angles of a triangle equals 180 degrees.

30 degrees + 80 degrees + x = 180 degrees

110 degrees + x = 180 degrees

x = 180 degrees − 110 degrees = 70 degrees

AnswerThe third angle is 70 degrees

Q5. The angles of a triangle are in the ratio 2:3:4. Find each angle.

Solution

Let the angles be 2x, 3x, and 4x.

By the angle sum property:

2x + 3x + 4x = 180 degrees

9x = 180 degrees

x = 20 degrees

The angles are:

2x = 2 × 20 = 40 degrees
3x = 3 × 20 = 60 degrees
4x = 4 × 20 = 80 degrees

Verification: 40 + 60 + 80 = 180 degrees (correct)

AnswerThe angles are 40 degrees, 60 degrees, and 80 degrees

Q6. One angle of a triangle is 90 degrees and another is 35 degrees. Find the third angle.

Solution

By the angle sum property:

90 degrees + 35 degrees + x = 180 degrees

125 degrees + x = 180 degrees

x = 180 degrees − 125 degrees = 55 degrees

AnswerThe third angle is 55 degrees

Exercise 4.2 — Congruence of Triangles

Q1. In isosceles triangle ABC with AB = AC, prove that angle B = angle C.

Solution

GivenTriangle ABC is isosceles with AB = AC.

To prove: Angle B = Angle C

Construction: Draw AD perpendicular to BC (AD &perp; BC), meeting BC at point D.

Proof:

In right-angled triangles ADB and ADC:

Angle ADB = Angle ADC = 90 degrees (since AD &perp; BC)
AD = AD (common side)
AB = AC (given, these are the hypotenuses)

By the RHS congruence rule (Right angle, Hypotenuse, Side):

Triangle ADB is congruent to Triangle ADC

Therefore, by CPCT (Corresponding Parts of Congruent Triangles):

Angle B = Angle C (proved)

Q2. In isosceles triangle ABC with AB = AC, AD is perpendicular to BC. Prove that D is the midpoint of BC.

Solution

GivenIn triangle ABC, AB = AC and AD &perp; BC.

To prove: D is the midpoint of BC (i.e., BD = CD)

Proof:

In triangles ABD and ACD:

Angle ADB = Angle ADC = 90 degrees (AD &perp; BC)
AD = AD (common side)
AB = AC (given, these are the hypotenuses)

By the RHS congruence rule:

Triangle ABD is congruent to Triangle ACD

Therefore, by CPCT:

BD = CD

Since BD = CD, point D is the midpoint of BC.

AnswerD is the midpoint of BC (proved)

Q3. PL &perp; OB, PM &perp; OA, PL = PM. Prove that Triangle PLO is congruent to Triangle PMO.

Solution

GivenPL &perp; OB, PM &perp; OA, and PL = PM.

To prove: Triangle PLO is congruent to Triangle PMO

Proof:

In right-angled triangles PLO and PMO:

Angle PLO = Angle PMO = 90 degrees (PL &perp; OB and PM &perp; OA)
OP = OP (common hypotenuse)
PL = PM (given)

By the RHS congruence rule (Right angle, Hypotenuse, Side):

Triangle PLO is congruent to Triangle PMO (proved)

Q4. In triangle ABC, AB = AC and angle B = 50 degrees. Find angle C.

Solution

GivenAB = AC and angle B = 50 degrees.

Since AB = AC, the angles opposite these equal sides are equal.

By the isosceles triangle property:

Angle B = Angle C (angles opposite equal sides are equal)

Therefore, angle C = 50 degrees.

AnswerAngle C = 50 degrees

Q5. In triangle PQR, PQ = PR and angle P = 40 degrees. Find angle Q and angle R.

Solution

GivenPQ = PR and angle P = 40 degrees.

Since PQ = PR, the angles opposite these equal sides are equal:

Angle Q = Angle R (angles opposite equal sides)

By the angle sum property:

Angle P + Angle Q + Angle R = 180 degrees

40 degrees + Angle Q + Angle R = 180 degrees

Since Angle Q = Angle R:

40 degrees + 2(Angle Q) = 180 degrees

2(Angle Q) = 140 degrees

Angle Q = 70 degrees

Therefore, Angle R = 70 degrees as well.

AnswerAngle Q = 70 degrees, Angle R = 70 degrees

Multiple Choice Questions (MCQs)

Q1. The sum of all angles of a triangle is:

(a) 90 degrees &emsp; (b) 360 degrees &emsp; (c) 180 degrees &emsp; (d) 270 degrees

Solution

By the angle sum property, the sum of all three interior angles of any triangle is 180 degrees.

Answer(c) 180 degrees

Q2. In an equilateral triangle, each angle equals:

(a) 60 degrees &emsp; (b) 90 degrees &emsp; (c) 120 degrees &emsp; (d) 45 degrees

Solution

In an equilateral triangle, all three sides are equal, so all three angles are equal.

Each angle = 180 degrees / 3 = 60 degrees.

Answer(a) 60 degrees

Q3. If two sides of a triangle are equal, the angles opposite them are:

(a) Complementary &emsp; (b) Equal &emsp; (c) Supplementary &emsp; (d) Right angles

Solution

By the isosceles triangle property, angles opposite equal sides are equal.

Answer(b) Equal

Q4. RHS congruence applies to:

(a) Any two triangles &emsp; (b) Equilateral triangles only &emsp; (c) Isosceles triangles only &emsp; (d) Right-angled triangles

Solution

RHS (Right angle, Hypotenuse, Side) congruence applies specifically to right-angled triangles.

Answer(d) Right-angled triangles

Q5. If one angle of a triangle is 110 degrees, the sum of the other two angles is:

(a) 90 degrees &emsp; (b) 70 degrees &emsp; (c) 110 degrees &emsp; (d) 180 degrees

Solution

Sum of all angles = 180 degrees.

If one angle = 110 degrees, the sum of the other two = 180 − 110 = 70 degrees.

Answer(b) 70 degrees

True or False

1. The sum of angles of a triangle can be more than 180 degrees.

AnswerFalse (the angle sum is always exactly 180 degrees for any triangle in Euclidean geometry)

2. An equilateral triangle is also an isosceles triangle.

AnswerTrue (an equilateral triangle has all three sides equal, which includes the condition that at least two sides are equal — the definition of isosceles)

3. Two right-angled triangles with equal hypotenuse and one equal side are congruent.

AnswerTrue (this is exactly the RHS congruence criterion: Right angle, Hypotenuse, Side)

4. In an isosceles triangle, the angle opposite the unequal side is called the vertex angle.

AnswerTrue (the unequal side is the base, and the angle opposite it — formed by the two equal sides — is the vertex angle)

5. Two triangles with equal areas are always congruent.

AnswerFalse (triangles can have equal areas but different shapes/sizes; congruence requires equal sides and angles, not just equal area)

Fill in the Blanks

1. The sum of angles of a triangle is ___

Answer180 degrees

2. In an isosceles triangle, angles opposite equal sides are ___

Answerequal

3. RHS stands for Right angle, Hypotenuse, ___

AnswerSide

4. If angle A = 50 degrees and angle B = 60 degrees in triangle ABC, then angle C = ___

Solutionangle C = 180 − 50 − 60 = 70 degrees

Answer70 degrees

5. An equilateral triangle has all angles equal to ___

Answer60 degrees

Common Mistakes

1. Using 360 degrees instead of 180 degrees for the angle sum of a triangle. A triangle always has an angle sum of 180 degrees. The 360 degrees applies to quadrilaterals.

2. Confusing the conditions for congruence. The valid congruence criteria are SSS, SAS, ASA, and RHS. SSA (Side-Side-Angle) is not a valid congruence criterion — this is a very common exam trap.

3. Forgetting that equal sides imply equal opposite angles (and vice versa) in isosceles triangles. If two sides are equal, the angles opposite them are equal. If two angles are equal, the sides opposite them are equal.

4. Assuming SSA is a valid congruence criterion. SSA is not a valid criterion. Two triangles with two sides and a non-included angle equal need not be congruent. Only RHS (which is a special case for right-angled triangles) works with this combination.

5. Not drawing auxiliary lines when proving congruence. When proving properties of isosceles triangles (like equal angles or midpoint), always draw an altitude or median to create two right-angled triangles for RHS congruence.

6. Mixing up CPCT (Corresponding Parts of Congruent Triangles). CPCT should only be applied after proving two triangles are congruent. You cannot use CPCT to prove congruence itself — that would be circular reasoning.

7. Confusing the vertex angle with base angles in an isosceles triangle. The vertex angle is the angle between the two equal sides. The base angles are the two equal angles opposite the equal sides. These are different angles.

Exam Checklist

Before your exam, make sure you can confidently:

[ ] Apply the angle sum property (angles of a triangle always sum to 180 degrees)
[ ] Classify triangles by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse)
[ ] Use the isosceles triangle property (equal sides imply equal opposite angles and vice versa)
[ ] State and apply all four congruence criteria (SSS, SAS, ASA, RHS)
[ ] Use CPCT correctly after proving congruence
[ ] Construct proofs involving isosceles triangles (drawing altitudes, using RHS)
[ ] Find unknown angles using the angle sum property and exterior angle theorem

Frequently Asked Questions

Q1: What is the angle sum property of a triangle?

Answer

The sum of the three interior angles of any triangle is always 180 degrees. This is true for all triangles — whether acute, right-angled, or obtuse. You can use this property to find any unknown angle when two angles of a triangle are known: third angle = 180 degrees minus the sum of the other two angles.

Q2: What does RHS congruence mean?

Answer

RHS stands for Right angle-Hypotenuse-Side. Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the corresponding parts of the other triangle. Note that RHS applies only to right-angled triangles. The right angle must be explicitly stated or proved before applying this criterion.

Q3: How do you prove that angles opposite equal sides are equal?

Answer

Draw an altitude from the vertex (the point where the equal sides meet) to the base. This altitude is common to both triangles. The two equal sides become the hypotenuses of two right-angled triangles. Since we have: a common side (the altitude), equal hypotenuses (the equal sides), and right angles, we can apply RHS congruence. The resulting triangles are congruent, so their base angles are equal by CPCT.

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