Learn to reason about and solve Grade 8 problems involving Congruent Figures.
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?
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?
Which sequence of rigid motions maps triangle ABC onto triangle DEF?
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?
Parallelogram MNOP is congruent to parallelogram QRST. In parallelogram MNOP, \(MN = 10\) cm and \(\angle M = 65°\). Which statement is true?
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?
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?
Which pair of figures is congruent?
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.
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.
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.
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.
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 \(\angle Q = 65°\).
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.
Congruent figures have equal corresponding sides. The congruent trapezoid has the same parallel sides: 8 cm and 12 cm.
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.
