Congruence of Triangles — Mcq
Congruence of Triangles — Lesson
1) Hook — A Fun Real-Life Story to Grab Your Attention
Imagine you are watching an intense cricket match between India and Australia. The bowler runs up and bowls a perfect delivery, and the batsman plays a shot that splits the fielders! Now, the coach wants to analyze the batsman’s stance and shot technique. He takes two photographs of the batsman’s stance from different angles but notices that the two images look exactly the same in shape and size. How can he be sure they are exactly the same? This is where the idea of congruence of triangles comes in — it helps us understand when two shapes are exactly identical in size and shape, just like those photos!
2) Core Concepts — Understanding Congruence of Triangles
What does Congruence mean?
Two triangles are said to be congruent if all their corresponding sides and angles are exactly equal. In simple words, one triangle can be placed on top of the other and they will match perfectly.
How to check if two triangles are congruent?
We use certain rules or criteria to check congruence without measuring every side and angle. These rules are based on matching parts of triangles.
| Congruence Rule | What it means | Example |
|---|---|---|
| SSS (Side-Side-Side) | All three sides of one triangle are equal to the three sides of another triangle. | If triangle ABC has sides 5 cm, 6 cm, 7 cm and triangle PQR has sides 5 cm, 6 cm, 7 cm, then △ABC ≅ △PQR. |
| SAS (Side-Angle-Side) | Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle. | If AB = PQ, AC = PR, and ∠A = ∠P, then △ABC ≅ △PQR. |
| ASA (Angle-Side-Angle) | Two angles and the included side of one triangle are equal to two angles and the included side of another triangle. | If ∠A = ∠P, ∠B = ∠Q, and AB = PQ, then △ABC ≅ △PQR. |
| AAS (Angle-Angle-Side) | Two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle. | If ∠A = ∠P, ∠B = ∠Q, and BC = QR, then △ABC ≅ △PQR. |
| RHS (Right angle-Hypotenuse-Side) | In right-angled triangles, if the hypotenuse and one side are equal, the triangles are congruent. | If right triangles ABC and PQR have hypotenuse AB = PQ and side AC = PR, then △ABC ≅ △PQR. |
Let’s look at a quick example:
Suppose two triangles have sides 3 cm, 4 cm, and 5 cm each. Using the SSS rule, these two triangles are congruent because all corresponding sides are equal.
3) Key Formulas / Rules
Congruence Criteria for Triangles:
- SSS: If three sides of one triangle are equal to three sides of another triangle, then triangles are congruent.
- SAS: If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, then triangles are congruent.
- ASA: If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, then triangles are congruent.
- AAS: If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, then triangles are congruent.
- RHS (for right triangles): If the hypotenuse and one side of a right triangle are equal to the corresponding parts of another right triangle, then triangles are congruent.
4) Did You Know?
In Indian architecture, especially in ancient temples like the Konark Sun Temple in Odisha, congruent triangles were used to create perfect symmetrical designs. These helped the temple maintain balance and beauty, much like how congruent triangles help in engineering and design today!
5) Exam Tips — Avoid These Common Mistakes!
- Don’t confuse similarity with congruence. Similar triangles have the same shape but not necessarily the same size; congruent triangles are identical in size and shape.
- Always match corresponding parts correctly. For example, side AB corresponds to side PQ, angle A corresponds to angle P, etc.
- Remember the order of letters in congruence statements. △ABC ≅ △PQR means A corresponds to P, B to Q, and C to R.
- Use the correct congruence rule. For example, don’t use ASA if the given parts are two sides and an angle not included between them.
- Practice drawing diagrams carefully. Visual clarity helps in understanding which parts correspond.
In the CBSE board exams, questions on congruence often ask you to prove two triangles are congruent using these rules or to find missing sides/angles using congruence. Make sure you write clear steps and mention the rule used!
Congruence of Triangles — Mnemonic
🎉 Memorable Mnemonics for Congruence of Triangles (Class 7) 🎉
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1. SAS - "Samosa And Samosa" 🍟🥟
Just like two samosas with the same Side and Angle filling are identical and tasty, two triangles with Side-Angle-Side equal are perfectly congruent! Remember: Samosa And Samosa = SAS.
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2. ASA - "Aloo Sabzi Always" 🥔🍲
Think of making aloo sabzi with the Angle of spices, then a Side of potatoes, and again an Angle of tadka – perfect recipe! Similarly, two triangles with Angle-Side-Angle equal are congruent. Aloo Sabzi Always = ASA.
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3. RHS - "Rohit Hits Sixes" 🏏💥
Imagine Rohit Sharma smashing sixes in cricket! When he hits a six, the Right angle of his bat swing, the Hypotenuse of his power, and the Side of his stance are perfect. Similarly, in right-angled triangles, Right angle-Hypotenuse-Side congruence means triangles are identical. Rohit Hits Sixes = RHS.
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