What does it mean?
Congruent shapes have the same size and shape. Moving, turning or reflecting a figure preserves its lengths and angles, so it remains congruent to the original. The vertices’ matching order identifies which sides and angles correspond, even if the drawings face different directions.
Two figures are congruent if one is the image of the other under a translation, reflection or rotation. Matching parts are corresponding. For triangles, SSS means three pairs of corresponding sides are equal. SAS means two sides and the included angle (the angle between those sides) are equal. The diagrams are not to scale. Use only named equal marks. SSA — two sides and a non-included angle — is not a congruence criterion taught here. AAA is about shape, not congruence of size.
Learning goal: Understand congruence through rigid transformations and apply triangle criteria. The grade is a suggested starting point; school sequences vary. Use the first worked example to check readiness, and Grade 7 triangle angles: missing and exterior angle questions if an earlier idea needs revisiting.
See the idea
Read each diagram with its labels, then follow the worked examples. Practice questions include their own measurements and information.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Choose a method and explain why it works
For triangles, match the givens against a complete criterion. SSS uses all three corresponding sides. SAS uses two corresponding sides and the included angle between them. State the criterion and the matching vertices. A combination of rigid movements may be needed to overlay the figures.
Notation: △ABC ≅ △DEF names corresponding vertices in order. Included angle is between the two given sides.
Key rule and when to use it
Congruent triangles match in shape and size. SSS compares three corresponding sides; SAS compares two sides and their included angle.
Worked examples, step by step
Example 1: Triangles ABC and DEF have AB = DE = 6, BC = EF = 7 and CA = FD = 8. Are they congruent? Name the criterion.
- Three pairs of sides are given equal. No extra unmarked sides are used.
- That is SSS. A rigid motion can overlay the triangles.
- So △ABC ≅ △DEF with A↔D, B↔E, C↔F.
Answer: yes; SSS
Check: Three matching side lengths determine a unique triangle size and shape.
Example 2: Triangles have sides 5 cm, 8 cm and included angle 40° matching. A second pair has sides 5 cm, 8 cm and a 40° angle that is not between those sides. Which pair is SAS?
- SAS needs the angle to sit between the two given sides.
- The first pair has the included 40°, so SAS applies.
- The second pair is SSA, which is not a congruence criterion here. Looking alike is not enough.
Answer: only the included-angle pair
Check: Moving the 40° to a different vertex changes the given information.
Example 3: Triangle ABC has sides 4, 6 and 7 cm. Triangle XYZ has matching sides 4, 6 and 7 cm. Does a rotated drawing change the conclusion?
- The three pairs of corresponding side lengths match.
- SSS proves the triangles congruent.
- A rotation changes position or orientation in the drawing, not the three lengths or the congruence.
Common mistakes and important boundaries
AAA establishes similarity, not equal size. Two sides and an arbitrary non-included angle do not generally establish congruence. The familiar nickname “diamond” is not a theorem, and an unmarked not-to-scale drawing cannot supply a missing equality. Use the given lengths and angle marks.
A point to explain
A learner may call triangles congruent because they look the same, or use SSA as if it were a criterion.
Compare an uncertain attempt with the closest worked example. Look at the meaning of each operation, unit and label before repeating the arithmetic.
Congruent Shapes and Matching Triangle Parts — practice
Work on paper first. Open each answer afterwards and compare the reasoning. Print the questions with or without the separate answer key; no account is required.
Name: ____________________ Date: __________
What does congruent mean for two triangles?
Answer and a way to check
They have the same size and shape, matching by a rigid motion.
Congruent means a slide, flip or turn can overlay the figures. Enlargement is not rigid.
All three corresponding side pairs of two triangles have equal lengths. Which congruence criterion applies?
Answer and a way to check
SSS
Count the marked sides. Three matching sides is SSS.
△PQR has PQ = 9, QR = 10, RP = 11. △XYZ has XY = 9, YZ = 10, ZX = 11. Name the criterion that proves them congruent.
Answer and a way to check
SSS
List the three side pairs. If all three match, the criterion is SSS.
Two triangles have sides 6 cm and 8 cm with the 50° angle between those sides marked equal. Name the criterion.
Answer and a way to check
SAS
The marked angle is between the two given sides. That is SAS.
Two triangles have two corresponding sides and an equal non-included angle. Is this information always enough to prove congruence? Answer yes or no.
Answer and a way to check
no
SAS needs the angle between the two sides. A different vertex is SSA, which is not a criterion here.
Two triangles look the same size on a not-to-scale sketch, but only two sides are marked equal. Can you conclude congruence? Name the rule you are using.
Answer and a way to check
No. Congruence needs a full criterion such as SSS or SAS from the marks, not appearance.
Read only the marks. Two equal sides without a third side or included angle is not SSS or SAS.
A student marks two sides and a non-included angle equal and writes SAS. What is the mistake?
Answer and a way to check
SAS requires the angle to be between the two sides.
Point to the angle. If it is not between the two given sides, it is not SAS.
A badge is a triangle with sides 5 cm, 7 cm, 9 cm. A second badge is the first after a 90° rotation. Give a congruence criterion and say whether a 2 times enlargement would still be congruent.
Answer and a way to check
SSS; enlargement is not congruent
A rotation is rigid, so all three sides stay 5, 7 and 9. Doubling lengths changes size.
Congruent Shapes and Matching Triangle Parts — answer key
They have the same size and shape, matching by a rigid motion. Congruent means a slide, flip or turn can overlay the figures. Enlargement is not rigid.
SSS Count the marked sides. Three matching sides is SSS.
SSS List the three side pairs. If all three match, the criterion is SSS.
SAS The marked angle is between the two given sides. That is SAS.
no SAS needs the angle between the two sides. A different vertex is SSA, which is not a criterion here.
No. Congruence needs a full criterion such as SSS or SAS from the marks, not appearance. Read only the marks. Two equal sides without a third side or included angle is not SSS or SAS.
SAS requires the angle to be between the two sides. Point to the angle. If it is not between the two given sides, it is not SAS.
SSS; enlargement is not congruent A rotation is rigid, so all three sides stay 5, 7 and 9. Doubling lengths changes size.
Questions students and parents ask
Does AAA prove congruence?
No. Matching angles establish similarity, but lengths may differ by a scale factor. Equilateral triangles of side 2 cm and 5 cm have the same angles without being congruent.
Are similar shapes always congruent?
No. Similarity allows a change of size. Similar figures with corresponding scale factor 1 are congruent, but a larger or smaller copy is not.
Is this a complete set of textbook exercise solutions?
No. This guide adapts BananaLearn’s own teaching examples and practice. It is not official NCERT material or a promise of the exact chapter sequence in a particular board, country or textbook edition.
What if the examples are still difficult?
Revisit Grade 7 triangle angles: missing and exterior angle questions. Try the first example with its model, then attempt one fresh question independently. A correct copied answer is not evidence that the method can be used without support.
Help at home
Ask the learner to list the precise givens before naming a criterion. This prevents a plausible-looking sketch from replacing a geometric argument.
Keep one example of the learner’s own working. A repeated difficulty across fresh questions can guide a discussion with their teacher; one short practice sheet does not establish a diagnosis or a full mastery profile.
Connect this idea to the next skill
About this guide
Based on BananaLearn’s Grade 8 course lesson “Same Size, Same Shape”. The explanations have been adapted for independent reading; course diagrams are reused where they fit the idea, and missing classroom givens are made explicit in questions.
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