Question 1

ΔABC is isosceles with AB = AC, and ∠A = 80°. What can we say about ∠B and ∠C?
Construct the altitude from A to BC.
Can you use this fact to find ∠B and ∠C?

Accepted Answer

Since ΔABC is isosceles with AB = AC, the angles opposite these sides are equal. Constructing the altitude AD from A to BC creates two right triangles, ΔADB and ΔADC. By RHS congruence (Right angle, Hypotenuse, Side), ΔADB ≅ ΔADC. Therefore, ∠B = ∠C. Since the sum of angles in ΔABC is 180°, we have ∠A + ∠B + ∠C = 180°. Given ∠A = 80°, and ∠B = ∠C, let each be x. Then 80° + x + x = 180°, so 2x = 100°, x = 50°. Hence, ∠B = ∠C = 50°. — 1. Given AB = AC, so ΔABC is isosceles.
2. Construct altitude AD perpendicular to BC.
3. Triangles ΔADB and ΔADC have:
   - AD common side,
   - AB = AC (given),
   - ∠ADB = ∠ADC = 90° (by construction).
4. By RHS congruence, ΔADB ≅ ΔADC.
5. Corresponding angles ∠B and ∠C are equal.
6. Sum of angles in ΔABC = 180°, so 80° + ∠B + ∠C = 180°.
7. Since ∠B = ∠C, 80° + 2∠B = 180° ⇒ 2∠B = 100° ⇒ ∠B = 50°.
8. Therefore, ∠B = ∠C = 50°.

Question 2

What can we say about the angles of an equilateral triangle?
Verify this by construction.
What could be their measures?

Accepted Answer

In an equilateral triangle, all sides are equal. Using the fact that angles opposite equal sides are equal, all three angles are equal. Since the sum of angles in a triangle is 180°, each angle measures 180° ÷ 3 = 60°. This can be verified by constructing an equilateral triangle and measuring its angles. — 1. Equilateral triangle has AB = BC = CA.
2. Angles opposite equal sides are equal, so ∠A = ∠B = ∠C.
3. Sum of angles in triangle = 180°.
4. Therefore, 3 × ∠A = 180° ⇒ ∠A = 60°.
5. Construction of equilateral triangle and measuring angles confirms this.

Question 3

ΔAIR ≅ ΔFLY. Identify the corresponding vertices, sides and angles.

2. Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent.

(a) AB = DE, BC = EF, CA = DF

(b) AB = EF, ∠A = ∠E, AC = ED

(c) AB = DF, ∠B = ∠D = 90°, AC = FE

(d) ∠A = ∠D, ∠B = ∠E, AC = DF

(e) AB = DF, ∠B = ∠F, AC = DE

3. It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. [Hint: AD is a transversal for these two lines. Are there any equal alternate angles?]

4. ABCD is a square. Show that ΔABC ≅ ΔADC. Is ΔABC also congruent to ΔCDA? Give more examples of two triangles where one triangle is congruent to the other in two different ways, as in the case above. Can you give an example of two triangles where one is congruent to the other in six different ways?

5. Find ∠B and ∠C, if A is the centre of the circle.

6. Find the missing angles. As per the convention that we have been following, all line segments marked with a single ‘|’ are equal to each other and those marked with a double ‘|’ are equal to each other, etc.

Accepted Answer

1. Corresponding vertices: A ↔ F, I ↔ L, R ↔ Y.
Corresponding sides: AI ↔ FL, IR ↔ LY, AR ↔ FY.
Corresponding angles: ∠A ↔ ∠F, ∠I ↔ ∠L, ∠R ↔ ∠Y.

2. (a) AB=DE, BC=EF, CA=DF → SSS congruence → ΔABC ≅ ΔDEF.
(b) AB=EF, ∠A=∠E, AC=ED → SAS congruence → ΔABC ≅ ΔEFD.
(c) AB=DF, ∠B=∠D=90°, AC=FE → RHS congruence → ΔABC ≅ ΔDFE.
(d) ∠A=∠D, ∠B=∠E, AC=DF → ASA congruence → ΔABC ≅ ΔDEF.
(e) AB=DF, ∠B=∠F, AC=DE → Not sufficient for congruence (two angles and non-included side).

3. Given OB=OC and OA=OD, consider transversal AD intersecting lines AB and CD. Since triangles OAB and OCD are congruent (by SSS), alternate interior angles are equal, so AB ∥ CD.

4. ABCD is a square. Triangles ΔABC and ΔADC share side AC, and AB=AD, BC=DC (all sides equal in square). By SAS, ΔABC ≅ ΔADC. Since ΔADC and ΔCDA are the same triangle (just different vertex order), ΔABC is not congruent to ΔCDA as distinct triangles.
Examples of triangles congruent in two different ways include isosceles right triangles where congruence can be shown by SAS and RHS.
Two triangles congruent in six different ways is not possible as there are only four standard congruence criteria.

5. Since A is the center of the circle, OA=OB=OC (radii). Triangle OBC is isosceles with OB=OC, so ∠B=∠C.
Using the properties of circle and triangle sum, find ∠B and ∠C accordingly.

6. Use the markings to identify equal sides and apply properties of triangles (sum of angles = 180°, angles opposite equal sides are equal) to find missing angles step-by-step. — Detailed reasoning for each part:

1. Correspondence is by naming order.

2. Use congruence criteria:
- SSS: all three sides equal.
- SAS: two sides and included angle equal.
- RHS: right angle, hypotenuse, and one side equal.
- ASA: two angles and included side equal.

3. Use congruent triangles and alternate interior angles to prove parallelism.

4. Use properties of square and congruence criteria.

5. Use circle radius equality and isosceles triangle properties.

6. Use markings and triangle angle sum to find missing angles.

Question 4

Draw lines and split the region consisting of white squares into 6 smaller congruent regions.

Accepted Answer

To split the given region of white squares into 6 smaller congruent regions, draw lines such that each smaller region is congruent in shape and size. This can be done by dividing the figure symmetrically, ensuring equal areas and congruent shapes. The exact lines depend on the figure shown, but typically involve drawing lines parallel or at angles that create equal partitions. — 1. Observe the shape and count the white squares.
2. Identify symmetry axes or points.
3. Draw lines dividing the region into 6 parts of equal area and shape.
4. Verify congruence by superimposition or measuring sides and angles.

Question 5

What does it mean when two geometric figures are said to be congruent?

Accepted Answer

Congruent figures are those that have the same shape and size. They can be exactly superimposed on each other, meaning one can be placed over the other so that they fit perfectly. — Congruent figures have identical shape and size, which means all corresponding sides and angles are equal. When placed one over the other, they coincide exactly without any gaps or overlaps. For example, two identical triangles or symbols are congruent if they match perfectly after rotation or flipping.

Question 6

If two symbols have arm lengths AB = 4 cm and BC = 8 cm, can these measurements alone guarantee that the two symbols are congruent?

Accepted Answer

No, arm lengths alone are not sufficient because the angle between them can vary — Having the same arm lengths AB and BC does not guarantee congruence because the angle ∠ABC can be different, resulting in different shapes. To ensure congruence, the angle between these arms must also be the same.

Question 7

Which three measurements are sufficient to recreate the symbol exactly if AB = 4 cm, BC = 8 cm, and the angle between them is 80°?

Accepted Answer

AB, BC, and ∠ABC — The two arm lengths AB and BC along with the included angle ∠ABC fix the shape and size of the figure uniquely, allowing exact reconstruction.

Question 8

Two figures are congruent if one can be exactly superimposed on the other. Which of the following transformations can be used before superimposing to check congruence?

Accepted Answer

Rotation or flipping (reflection) — Congruence allows rotation and flipping of figures before superimposing. Scaling or shearing changes size or shape, so they are not allowed for congruence.

Question 9

How can you check if two circles are congruent?

Accepted Answer

Two circles are congruent if they have the same radius. For example, measuring the radius of both circles and confirming they are equal ensures congruence. — Since all circles with the same radius have the same size and shape, equal radii guarantee congruence. Circles can be superimposed exactly by aligning their centers.

Question 10

What measurements would you take to check if two rectangles are congruent?

Accepted Answer

Two rectangles are congruent if their corresponding lengths and widths are equal. For example, measuring the length and width of both rectangles and confirming they are equal ensures congruence. — Rectangles have opposite sides equal and all angles 90°. If two rectangles have the same length and width, they have the same shape and size, hence are congruent.

Question 11

Meera and Rabia measured the sides of a large triangular frame as 40 cm, 60 cm, and 80 cm. Meera says that these side lengths are sufficient to construct a congruent triangle without measuring angles. Do you agree? Justify your answer.

Accepted Answer

Yes, I agree with Meera because the SSS (Side Side Side) condition states that if two triangles have the same three side lengths, then they are congruent. So, measuring the three sides is sufficient to construct a congruent triangle. — The SSS condition guarantees congruence of triangles. Since the three sides are fixed, the shape and size of the triangle are fixed, and angles are automatically determined. Therefore, measuring angles is not necessary.

Question 12

Rabia constructs two triangles ABE and ABF with sides AB = 6 cm, AE = 4 cm, and BF = 8 cm using circles from endpoints of AB. Are these two triangles congruent? Explain why or why not.

Accepted Answer

Yes, the two triangles ABE and ABF are congruent. They are mirror images of each other across the line AB, and have the same three side lengths, satisfying the SSS condition. — Both triangles share side AB, and the other two sides are radii of circles drawn from A and B with lengths 4 cm and 8 cm respectively. The two intersection points create two congruent triangles symmetric about AB.

Question 13

Which of the following notations correctly expresses the congruence of triangles with corresponding vertices A, B, C and X, Y, Z respectively?

Accepted Answer

ΔABC ≅ ΔXYZ — The notation ΔABC ≅ ΔXYZ means vertex A corresponds to X, B to Y, and C to Z, ensuring corresponding sides and angles overlap. The other notations mismatch vertex order or correspondence.

Question 14

In rectangle ABCD, triangles ABD and CDB are formed. Given AB = CD, AD = CB, and BD is common, are these triangles congruent? If yes, state the condition used.

Accepted Answer

Yes, triangles ABD and CDB are congruent by the SSS condition because all three sides AB = CD, AD = CB, and BD is common to both triangles. — Since all three pairs of corresponding sides are equal, the triangles satisfy the SSS condition of congruence and hence are congruent.

Question 15

Identify the correct correspondence of vertices to express the congruence between triangles ABD and CDB in rectangle ABCD.

Accepted Answer

A corresponds to D, B to C, D to B — For the triangles to be congruent, corresponding sides must be equal. The correct vertex correspondence is A with D, B with C, and D with B, ensuring equal sides overlap.

Geometric

Geometric — Study Notes

1.1 Geometric Twins

1.2 Congruence of Triangles

1.3 Angles of Isosceles and Equilateral Triangles

Practice Questions — Geometric

All 7 Chapters in Ganita Prakash-II