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Geometry Worksheet Solution: Triangle Congruence and Proofs
Published on October 4, 2026
Geometry
Step-by-step solutions for a grade 7 mathematics worksheet on geometric twins, triangle congruence criteria, and proofs involving isosceles triangles.
Question
Solve with explanation
Answer
Here are the step-by-step solutions for the provided mathematics worksheet:
1. Figures that have the same shape and size are called ________.
- Reasoning: By definition, figures that are identical in both shape and size are called congruent. Similar figures have the same shape but not necessarily the same size.
- Answer: © congruent
2. Which of the following symbols is used to denote congruence?
- Reasoning: The symbol
is the standard mathematical notation for congruence. The symbol is used for similarity. - Answer: (d)
3. If
- Reasoning: In congruent triangles, corresponding parts (angles and sides) are equal (CPCT).
-
corresponds to (Correct) -
corresponds to (Incorrect; the option says ) -
corresponds to (Correct) -
corresponds to (Correct)
-
- Answer: (b)
4. Assertion (A): If two triangles are congruent, their corresponding sides are equal. Reason ®: Congruent triangles have the same shape and size.
- Reasoning: Assertion (A) is true because congruence implies equality of all corresponding parts. Reason ® is also true as it is the definition of congruence. Since having the same size and shape is exactly why the corresponding sides must be equal, ® is the correct explanation for (A).
- Answer: (a) Both assertion (A) and reason ® are true and reason ® is the correct explanation of assertion (A).
5. Two right-angled triangles
- Step 1: Identify given parts:
(Right angle), (Hypotenuse), and (One side). - Step 2: According to the RHS (Right angle-Hypotenuse-Side) congruence rule, if the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, they are congruent.
- Answer: Yes, the triangles are congruent by the RHS congruence rule.
6. Using congruence of triangles, prove that the base angles of an isosceles triangle are equal.
- Step 1: Let
be an isosceles triangle where . Draw a bisector of that meets at point . - Step 2: In
and : -
(Given) -
(By construction of angle bisector) -
(Common side)
-
- Step 3: Therefore,
by the SAS congruence rule. - Step 4: By CPCT (Corresponding Parts of Congruent Triangles),
. - Conclusion: The base angles of an isosceles triangle are equal.
7. Can two triangles be congruent by AAA criteria? Why or why not?
- Answer: No.
- Reasoning: The AAA (Angle-Angle-Angle) criterion only proves that two triangles are similar, meaning they have the same shape. It does not guarantee they have the same size. For example, a small equilateral triangle and a large equilateral triangle both have angles of
, but they are not congruent because their side lengths differ.
8.
- Step 1: Since
, is an isosceles triangle. Therefore, (Base angles of an isosceles triangle are equal). - Step 2: Given
. When a transversal ( ) intersects parallel lines, corresponding angles are equal. Thus, . - Step 3: Similarly, for transversal
, . - Step 4: Since
, it follows that . - Step 5: In
, since the base angles and are equal, the opposite sides must be equal. - Conclusion:
.