NCERT Solutions for Class 7th Maths Chapter 1 In-text Questions — Angles of Isosceles and Equilateral Triangles · Congruent Triangles in Real Life
Book page 17–20 Updated on2026-09-19
Q1.
ΔABC is isosceles with AB = AC, and ∠A = 80°. What can we say about ∠B and ∠C?
Answer
∠B = ∠C. The angles opposite the two equal sides are equal.
Drop the altitude from A to BC, meeting BC at D. Then compare ∆ADB and ∆ADC:
AB = AC (given) ∠ADB = ∠ADC = 90° (AD is the altitude) AD = AD (common side) → RHS condition → ∆ADB ≅ ∆ADC
So the corresponding angles ∠B and ∠C are equal.
Why it happens: AB and AC are the hypotenuses of the two small right triangles, and they are equal. AD is common. That is the right angle, the hypotenuse and one more side — exactly RHS. This is a general result: in a triangle, angles opposite to equal sides are equal.
Q2.
Can you use this fact to find ∠B and ∠C?
Answer
Yes. Call each of them x.
∠A + ∠B + ∠C = 180° 80° + x + x = 180° 2x = 100° x = 50° ∠B = ∠C = 50°
Why it happens: once we know the two base angles are equal, the angle sum turns into a single equation with one unknown. Whatever is left after taking 80° out of 180° has to be shared equally between ∠B and ∠C.
Check it yourself: 80° + 50° + 50° = 180° ✓
Q3.
Equilateral triangles are those in which all the sides have equal lengths. What can we say about their angles?
Answer
All three angles are equal.
Use the fact just discovered, twice over, in ∆ABC:
AB = AC → the angles opposite them are equal → ∠C = ∠B AB = BC → the angles opposite them are equal → ∠C = ∠A → ∠A = ∠B = ∠C
Why it happens: "angles opposite equal sides are equal" can be applied to any pair of equal sides. An equilateral triangle has three such pairs, so all three angles get tied together. Just like the sides, the angles are all alike.
Q4.
What could be their measures?
Answer
Each angle is 60°.
∠A + ∠B + ∠C = 180° All three are equal, so 3 × (angle of an equilateral triangle) = 180° angle = 180° ÷ 3 = 60°
Why it happens: the angle sum of a triangle is fixed at 180°, and here it has to be divided into three equal shares. Notice that we never measured anything — congruence alone was enough to prove that every equilateral triangle in the world has three 60° angles.
Q5.
Verify this by construction.
Answer
Construct an equilateral triangle and measure its angles.
Draw AB = 5 cm.
With A as centre and radius 5 cm, draw an arc above AB.
With B as centre and the same radius 5 cm, draw another arc cutting the first at C.
Check it yourself: repeat with a side of 3 cm and again with 8 cm. The triangles are of different sizes, but every angle still measures 60°. The size of an equilateral triangle does not change its angles.
Q6.
Congruent Triangles in Real Life: Congruent triangles can be seen in various constructions and designs from ancient to modern times. Describe the congruent triangles you see in each picture. (Louvre Museum, Pyramid of Giza, dome design, rangoli design, Rabindra Setu or Howrah Bridge)
Answer
Congruent triangles appear in every one of the pictures.
Picture
Congruent triangles you can see
Why they are congruent
Louvre Museum, Paris
The glass pyramid is covered with rows of small triangular panes
Each pane is cut to the same size, so all panes in a row are copies of one another
Pyramid of Giza
The four slanting faces of the pyramid
The base is a square and the apex is directly above its centre, so the four faces have the same sidelengths
Dome design
Triangles repeating around the curved surface
The same triangular piece is repeated after turning it through a fixed angle
Rangoli design
The triangular petals around the centre
The pattern is made by turning one triangle again and again about the centre point
Rabindra Setu (Howrah Bridge)
The steel triangles in the criss-cross girders
The bars are cut to fixed lengths, so each triangle in the truss repeats along the bridge
Why it happens: repeating one congruent piece is cheap and strong. A builder needs only one measurement set and one mould, and a triangle cannot be pushed out of shape the way a rectangle can — its three sides fix it completely, by SSS. That is why bridges and domes are built out of triangles.
Try This: look for congruent triangles around you — a folding chair, a kite, an ironing board, or the pattern on a dupatta.