CBSE Class 7 Mathematics Congruence Of Triangles Worksheet Set 03

Chapter 7: Congruence of Triangles

Question 1. Define congruence of triangles.
Answer: Two triangles are congruent if they have the exact same shape and the exact same size. When one triangle is placed over the other, all three matching sides and all three matching angles cover each other completely.
In simple words: Congruent triangles are twin shapes. They have the same side lengths and angle measures.

Exam Tip: Always include both keywords "same shape" and "same size" in your definition to secure full marks.

Question 2. Write criteria of congruence of a triangle.
Answer: The four standard criteria for congruence of triangles are:
1. SSS (Side-Side-Side) Criterion: All three sides of one triangle are equal to the three corresponding sides of another triangle.
2. SAS (Side-Angle-Side) Criterion: Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.
3. ASA (Angle-Side-Angle) Criterion: Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
4. RHS (Right angle-Hypotenuse-Side) Criterion: In two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.
In simple words: These are the four rules - SSS, SAS, ASA, and RHS - used to prove two triangles are identical.

Exam Tip: Write down both the abbreviation and its full form (e.g., SAS - Side-Angle-Side) when listing congruence criteria.

Question 3. Which congruence criteria do you use in the above figure.
AB = PQ ; Angle B = Angle Q ; BC = QR
Answer: The congruence criterion used in the given figure is SAS (Side-Angle-Side).
In \( \Delta \text{ABC} \) and \( \Delta \text{PQR} \):
- \( \text{AB} = \text{PQ} \) (Side)
- \( \angle \text{B} = \angle \text{Q} \) (Included Angle)
- \( \text{BC} = \text{QR} \) (Side)
Since two sides and the angle between them in \( \Delta \text{ABC} \) are equal to the matching two sides and the angle between them in \( \Delta \text{PQR} \), the two triangles are congruent.
Therefore, \( \Delta \text{ABC} \cong \Delta \text{PQR} \) by the SAS criterion.
In simple words: Two sides and the corner between them match in both shapes, so SAS is the correct rule.

Exam Tip: Confirm that the given angle is the "included angle" located between the two marked sides before applying the SAS rule.

Question 4. Give any three real life example for congruent shape.
Answer: Three everyday examples of congruent shapes are:
1. Sheets of paper taken from the same unused notebook or newly opened A4 ream.
2. Biscuits packed inside the same fresh packet.
3. Two one-rupee coins minted in the exact same year.
In simple words: Identical objects from the same pack, like notebook pages or coins of the same year, are congruent shapes.

Exam Tip: When mentioning coins or stamps, state that they must be minted or printed in the same year or series to ensure identical size.

Question 5. In
AB = 5 cm ; angle B = 50⁰ ; BC = 5.5 cm ; & DF = 5 cm angle F = 50⁰ & FE = 5.5 cm , then write the triangles in order they are congruent & Write also criteria.
Answer: In \( \Delta \text{ABC} \) and \( \Delta \text{DFE} \):
- \( \text{AB} = \text{DF} = 5\text{ cm} \) (Side)
- \( \angle \text{B} = \angle \text{F} = 50^\circ \) (Included Angle)
- \( \text{BC} = \text{FE} = 5.5\text{ cm} \) (Side)
The corresponding vertices match as follows: vertex A corresponds to D, vertex B corresponds to F, and vertex C corresponds to E.
Therefore, the triangles written in congruent order are:
\( \Delta \text{ABC} \cong \Delta \text{DFE} \)
Congruence criterion: SAS (Side-Angle-Side) Congruence Criterion.
In simple words: The 50-degree angle sits between the 5 cm and 5.5 cm sides in both shapes. Matching each letter correctly gives triangle ABC congruent to triangle DFE by SAS.

Exam Tip: Be careful with letter order in congruence statements. Vertex B must match vertex F because both have the 50-degree angle.

Question 6. In the figure , AB = AC & AD is the bisector of angle BAC.
(i) State three pairs of equal parts in triangles ADB & ADC.
(ii) Is triangle ADB is congruent to triangle ADC .
Answer:
(i) In \( \Delta \text{ADB} \) and \( \Delta \text{ADC} \), the three pairs of equal parts are:
- \( \text{AB} = \text{AC} \) (Given)
- \( \angle \text{BAD} = \angle \text{CAD} \) (Given that AD bisects \( \angle \text{BAC} \))
- \( \text{AD} = \text{AD} \) (Common side shared by both triangles)
(ii) Yes, triangle ADB is congruent to triangle ADC (\( \Delta \text{ADB} \cong \Delta \text{ADC} \)).
Reason: By SAS (Side-Angle-Side) congruence rule, two sides and their included angle are equal in both triangles.
In simple words: The outer sides are equal, the top angle is cut into two equal halves, and AD is a shared side. By the SAS rule, both triangles are congruent.

Exam Tip: Since AD bisects angle BAC, the two smaller angles at vertex A are equal. State this clearly as your reason.

Question 7. Which angle is included between DE & EF of triangle DEF?
Answer: In \( \Delta \text{DEF} \), the sides DE and EF meet at the common vertex E.
Therefore, the angle included between sides DE and EF is \( \angle \text{E} \) (or \( \angle \text{DEF} \)).
In simple words: Both sides share the letter E, so angle E is the corner formed between them.

Exam Tip: To find the included angle between two sides quickly, look for the letter that appears in both side names.

Question 8. In a squared sheet , draw two triangles of equal areas such that
(i) the triangle are congruent .
(ii) the triangles are not congruent .
What can you say about their perimeter ?
Answer:
(i) Congruent triangles of equal area:
Draw two right-angled triangles each having base = \( 4\text{ units} \) and height = \( 3\text{ units} \).
Area of each triangle = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6\text{ sq units} \).
Since all corresponding sides are equal (\( 3\text{ units} \), \( 4\text{ units} \), and hypotenuse \( 5\text{ units} \)), the triangles are congruent.
Perimeter of each triangle = \( 3 + 4 + 5 = 12\text{ units} \).
(ii) Non-congruent triangles of equal area:
Draw a right-angled triangle with base = \( 4\text{ units} \) and height = \( 3\text{ units} \), so Area = \( \frac{1}{2} \times 4 \times 3 = 6\text{ sq units} \).
Draw another right-angled triangle with base = \( 6\text{ units} \) and height = \( 2\text{ units} \), so Area = \( \frac{1}{2} \times 6 \times 2 = 6\text{ sq units} \).
Both triangles have the same area of \( 6\text{ sq units} \), but their sides are different, so they are not congruent.
Perimeter of the first triangle = \( 3 + 4 + 5 = 12\text{ units} \).
Perimeter of the second triangle = \( 6 + 2 + \sqrt{6^2 + 2^2} = 8 + \sqrt{40} \approx 14.32\text{ units} \).
Conclusion about their perimeter:
- When two triangles are congruent, their perimeters are always equal.
- When two triangles are not congruent, their perimeters are generally not equal.
In simple words: Congruent triangles of the same area always have equal perimeters. But non-congruent triangles of the same area can have completely different perimeters.

Exam Tip: Remember that having equal areas does not mean triangles are congruent; their boundary lengths can still differ.

Question 9. If AC = DC ; angle ABC = angle DBC & BC = BC
STATE THE CRITERIA OF THE CONGRUENCE.
Answer: In the given figure, segment AB and segment DB are perpendicular to BC, so \( \angle \text{ABC} = \angle \text{DBC} = 90^\circ \).
In right triangles \( \Delta \text{ABC} \) and \( \Delta \text{DBC} \):
- \( \angle \text{ABC} = \angle \text{DBC} = 90^\circ \) (Right angle)
- \( \text{AC} = \text{DC} \) (Hypotenuse)
- \( \text{BC} = \text{BC} \) (Common side)
Therefore, the congruence criterion is the RHS (Right angle-Hypotenuse-Side) criterion.
Thus, \( \Delta \text{ABC} \cong \Delta \text{DBC} \) by RHS congruence rule.
In simple words: Both triangles share a 90-degree angle, equal hypotenuses, and the common side BC. This matches the RHS rule.

Exam Tip: When the hypotenuse and one leg of two right triangles are equal, always name RHS as the criterion.

Question 10. If DA is perpendicular to AB & CB is perpendicular to BA & AC = BD
Then state which two triangle are congruent & by which criteria.
Answer: In \( \Delta \text{DAB} \) and \( \Delta \text{CBA} \):
- \( \angle \text{DAB} = \angle \text{CBA} = 90^\circ \) (Given that \( \text{DA} \bot \text{AB} \) and \( \text{CB} \bot \text{BA} \))
- \( \text{BD} = \text{AC} \) (Given hypotenuses are equal)
- \( \text{AB} = \text{BA} \) (Common side)
Therefore, the two congruent triangles are:
\( \Delta \text{DAB} \cong \Delta \text{CBA} \) (or \( \Delta \text{ABD} \cong \Delta \text{BAC} \))
Congruence criterion: RHS (Right angle-Hypotenuse-Side) Congruence Criterion.
In simple words: Triangles DAB and CBA share the base AB, have right angles at the bottom, and have matching diagonal hypotenuses. By RHS, triangle DAB is congruent to triangle CBA.

Exam Tip: Match the corresponding vertices properly: vertex D corresponds to C, A corresponds to B, and B corresponds to A.