📐 Geometry

Explore the properties of similar triangles

In the given △ABC ~ △PQR, then

a) Write down the name of corresponding angles and sides.

b) Find the value of ∠PRQ.

Solution

Here, △ABC ~ △PQR.

Step 1 — Given △ABC ~ △PQR, so vertices correspond in order:
A ↔ P, B ↔ Q, C ↔ R

Step 2 — Corresponding Angles (part a) Since A↔P, B↔Q, C↔R:
∠A and ∠P, ∠B and ∠Q, ∠C and ∠R

Step 3 — Corresponding Sides (part a) Sides opposite corresponding vertices:
BC and QR, AC and PR, AB and PQ

Step 4 — Find ∠PRQ (part b) ∠ACB = ∠PRQ (Corresponding angles are equal of similar triangles)

∴ ∠PRQ = 50°

✅ Answer Therefore: ∠BCA = ∠PRQ = 50°
Thus, ∠PRQ = 50°

If △PQR ~ △XYZ in the adjoining figure, answer the following questions.

a) Find the ratio of corresponding sides.

b) Find the value of ∠PQR.

c) Find the length (measurement) of side PR.

Solution

We're given that △PQR ~ △XYZ. So the corresponding vertices are:
P ↔ X, Q ↔ Y, R ↔ Z

Step 1 — Ratio of Corresponding Sides (part a) From the figure: QR = 8 cm, YZ = 16 cm

QRYZ = 816 = 12

Thus, the ratio of corresponding sides is: 1 : 2

Step 2 — Find ∠PQR (part b) Using similarity: ∠PQR = ∠XYZ
From the figure: ∠XYZ = 75°

∴ ∠PQR = 75°

Step 3 — Find Length of PR (part c) Corresponding sides: PR ↔ XZ
Scale factor = 12

Side XY = 8 cm corresponds to PQ. Using the ratio:

PRXZ = 12

PR = 12 × 8 = 4 cm

✅ Answer a) Ratio of corresponding sides = 1 : 2
b) ∠PQR = 75°
c) PR = 4 cm

Question 4 — Find the values of x and y from the given similar triangles.

Two similar triangles have angles (5y−3)°, 70°, (4x−4)° and 70°, 62°, 48°. Find the values of x and y.

Solution

Step 1 — Identify Corresponding Angles Triangles are similar → corresponding angles are equal.
Angles in second triangle: 70°, 62°, 48°

Match with first triangle:
70° ↔ 70°, (5y−3)° ↔ 62°, (4x−4)° ↔ 48°

Step 2 — Solve for y

5y − 3 = 62°	(corresponding angles of similar triangles are equal)
5y = 62° + 3
5y = 65°
y = 65°5 = 13°

Step 3 — Solve for x

4x − 4 = 48°	(corresponding angles of similar triangles are equal)
4x = 48° + 4
4x = 52°
x = 52°4 = 13°

✅ Answer x = 13° and y = 13°

Question 6 — Find the value of x (△ABC ~ △PQR).

△ABC ~ △PQR. Sides: AB = 3.1 cm, AC = (3x−0.6) cm in △ABC; PQ = 3.1 cm, PR = (x+3) cm in △PQR. Angles: 56° and 64°. Find x.

Solution

Step 1 — Identify Corresponding Sides △ABC ~ △PQR
corresponding sides: AB ↔ PQ, AC ↔ PR

Set up proportion:
ABPQ = ACPR (corresponding sides of similar triangle are proportional)

3.13.1 = 3x − 0.6x + 3

Step 2 — Simplify Left Side 3.13.1 = 1

So: 1 = 3x − 0.6x + 3

Step 3 — Cross Multiply 1 × (x + 3) = 3x − 0.6
x + 3 = 3x − 0.6

Step 4 — Solve for x 3 + 0.6 = 3x − x
3.6 = 2x

x = 3.62 = 1.8

✅ Answer x = 1.8

PR = x + 3 = 1.8 + 3 = 4.8 cm
AC = 3x − 0.6 = 3(1.8) − 0.6 = 5.4 − 0.6 = 4.8 cm

Solve — In △ABC ~ △PQR, find ∠PRQ.

Given: △ABC ~ △PQR where ∠BAC = 55°, ∠ABC = 75°, ∠BCA = 50°. Find the value of ∠PRQ.

📝 Solve Step by Step — Choose the correct answer for each step:

🎉 Excellent!

You have mastered the properties of similar triangles.

In a similar triangle, equal angles are called corresponding angles and opposite sides of equal angles are called corresponding sides. When the corresponding angles of two triangles are equal and the corresponding sides are proportional, then those two triangles are called similar triangles.

Similar triangles are denoted by △ABC ~ △PQR.