Introduction
Congruent Figures is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with congruent figures.
What Is Congruent Figures?
Congruent Figures means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Congruent Figures
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Two congruent quadrilaterals are shown. Quadrilateral ABCD corresponds to quadrilateral EFGH in order (A\(rightarrow\)E, B\(rightarrow\)F, C\(rightarrow\)G, D\(rightarrow\)H). Which statement must be true?
- A. Side \(AB\) is longer than side \(EF\)
- B. Angle \(A\) equals angle \(E\)
- C. Side \(CD\) is shorter than side \(GH\)
- D. The perimeters are different
Why it works: From the diagram, quadrilateral ABCD (left) is congruent to EFGH (right) with vertices labeled in order: A→E, B→F, C→G, D→H. The tick marks on sides confirm this correspondence. Since congruent figures have equal corresponding angles, \(\angle A = \angle E\), \(\angle B = \angle F\), and so on.
Answer: Angle \(A\) equals angle \(E\)
Visual Model 2
Question: Two congruent rectangles are shown. Rectangle 1 has a length of 12 inches and a width of 5 inches. What is the perimeter of Rectangle 2?
- A. 17 inches
- B. 24 inches
- C. 34 inches
- D. 60 inches
Why it works: The perimeter of Rectangle 1 is \(2(12) + 2(5) = 24 + 10 = 34\) inches. Since Rectangle 2 is congruent, it has the same perimeter: 34 inches.
Answer: 34 inches
Worked Examples
Example 1
Question: Which sequence of rigid motions maps triangle ABC onto triangle DEF?
- A. Translation only
- B. Reflection and translation
- C. Rotation only
- D. Rotation and reflection
- Comparing the triangles: Triangle ABC has vertex A at the bottom left, B at the bottom right, and C at the top.
- Triangle DEF has vertices in a different orientation with D at the bottom left, E at the bottom right, and F at the bottom (flipped).
- A reflection flips triangle ABC, and then a translation (shift) moves it to the position of triangle DEF.
Answer: Reflection and translation
Example 2
Question: Two congruent isosceles triangles are shown with tick marks indicating equal sides. Triangle 1 has a base of 6 cm and equal sides of 5 cm each. What is the perimeter of Triangle 2?
- A. 11 cm
- B. 16 cm
- C. 22 cm
- D. 30 cm
- Perimeter of Triangle 1 = \(5 + 5 + 6 = 16\) cm.
- Even though Triangle 2 is oriented differently (rotated or reflected), congruent figures have equal corresponding sides.
- Therefore, Triangle 2 also has sides of 5 cm, 5 cm, and 6 cm, giving a perimeter of 16 cm.
Answer: 16 cm
Example 3
Question: Parallelogram MNOP is congruent to parallelogram QRST. In parallelogram MNOP, \(MN = 10\) cm and \(\angle M = 65°\). Which statement is true?
- A. \(QR = 10\) cm and \(\angle Q = 115°\)
- B. \(QR = 10\) cm and \(\angle Q = 65°\)
- C. \(QR = 20\) cm and \(\angle Q = 65°\)
- D. \(QR\) cannot be determined
- Congruent parallelograms have equal corresponding sides and angles.
- Side \(MN\) corresponds to side \(QR\), so \(QR = 10\) cm.
- Angle \(M\) corresponds to angle \(Q\), so \(\angle Q = 65°\).
Answer: \(QR = 10\) cm and \(\angle Q = 65°\)
Real-World Word Problems
Problem 1
Question: Two congruent regular polygons are shown. Each polygon has 8 sides. If one octagon has a side length of 3 inches, what is the side length of the other octagon?
- A. 1.5 inches
- B. 3 inches
- C. 6 inches
- D. 9 inches
Why it works: Congruent regular polygons have equal side lengths. Both octagons have side length 3 inches.
Answer: 3 inches
Problem 2
Question: Two congruent trapezoids are shown with side length markings. Trapezoid 1 has a height of 4 inches. What is the height of Trapezoid 2?
- A. 2 inches
- B. 4 inches
- C. 8 inches
- D. Cannot be determined
Why it works: Congruent trapezoids have equal heights. Since Trapezoid 1 has a height of 4 inches and Trapezoid 2 is congruent, Trapezoid 2 also has a height of 4 inches.
Answer: 4 inches
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Two triangles are congruent. If one triangle has angles of \(40°, 60°, 80°\), what are the angles of the other triangle?
- A. \(40°, 60°, 90°\)
- B. \(40°, 60°, 80°\)
- C. \(80°, 60°, 40°\) (different order)
- D. \(50°, 60°, 70°\)
Question 2
Pentagon PQRST is congruent to pentagon UVWXY. If side \(PQ = 8\) cm, what is the length of the corresponding side in the second pentagon?
- A. 4 cm
- B. 8 cm
- C. 16 cm
- D. Cannot be determined
Question 3
Hexagon ABCDEF is congruent to hexagon GHIJKL. In hexagon ABCDEF, the interior angles are: \(120°, 120°, 120°, 120°, 120°, 120°\). Which of the following must be true about hexagon GHIJKL?
- A. All its angles are 90°
- B. All its angles are 120°
- C. Its angles sum to 360°
- D. Some of its angles are greater than 120°
Question 4
Triangle XYZ is mapped onto triangle X'Y'Z' by a 90-degree counterclockwise rotation about the origin. If the triangles are congruent, which of the following is true?
- A. Side \(XY\) has a different length than side X'Y'
- B. The angle at Z differs from the angle at Z'
- C. Side \(XZ =\) side X'Z'
- D. All angles are different after rotation
Question 5
Two congruent trapezoids are shown. One trapezoid has parallel sides of 8 cm and 12 cm. What are the parallel sides of the congruent trapezoid?
- A. 4 cm and 6 cm
- B. 8 cm and 12 cm
- C. 16 cm and 24 cm
- D. Cannot be determined
Question 6
Which pair of figures is congruent?
- A. Figures A and B
- B. Figures A and C
- C. Figures B and C
- D. None are congruent
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(40°, 60°, 80°\)
Congruent triangles have equal corresponding angles. Regardless of which triangle is labeled first or second, the three angles must be identical: \(40°, 60°, 80°\). The order of listing doesn't matter; the angles are the same.
Question 2
Answer: 8 cm
Congruent figures have equal corresponding sides. The side corresponding to \(PQ\) in pentagon UVWXY is \(UV\), and it must equal 8 cm.
Question 3
Answer: All its angles are 120°
Since the hexagons are congruent, all corresponding angles are equal. Hexagon GHIJKL has the same angles as ABCDEF: \(120°, 120°, 120°, 120°, 120°, 120°\). (Note: The interior angle sum of any hexagon is always 720°, so option C is incorrect.)
Question 4
Answer: Side \(XZ =\) side X'Z'
A 90-degree counterclockwise rotation is a rigid motion that preserves all distances and angles. When triangle XYZ is rotated about the origin, each vertex moves but distances between vertices remain the same. Therefore, all corresponding sides and angles are equal: \(XZ = X'Z'\), \(XY = X'Y'\), \(YZ = Y'Z'\), and all angles match.
Question 5
Answer: 8 cm and 12 cm
Congruent figures have equal corresponding sides. The congruent trapezoid has the same parallel sides: 8 cm and 12 cm.
Question 6
Answer: Figures A and C
Figures A and C are both \(4 \times 2\) rectangles. Figure B is a \(2 \times 2\) square, so it is not congruent to either rectangle.
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Congruent Figures becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
