Congruent Figures
What does it mean for two figures to be congruent?
Two figures are congruent if they both have the same shape and size.
Two squares are similar. The side length of the first square is 5 cm, and the side length of the second square is 10 cm. What is the scale factor between the two squares?
The scale factor is the ratio of the side lengths.
10 cm / 5 cm = 2
So, the scale factor between the two squares is 2.
In a right triangle, if the angle is 90° and the hypotenuse is 5 cm, what is the length of the side opposite the 90° angle?
In any right triangle, the side opposite the 90° angle is always the hypotenuse. So, the side opposite the 90° angle is 5 cm.
Are a square and rectangle always congruent?
No, because a square has all sides the same length, but a rectangle can have different side lengths. They are only congruent if all their sides and angles are exactly the same.
Are two shapes with the same shape but different sizes always similar?
Yes, two shapes with the same shape but different sizes are always similar.
In triangle ABC, angle A is 30°, angle B is 45°, and side a (opposite angle A) is 6 cm. What is the length of side b (opposite angle B)?
Side b is approximately 8.49 cm
If two triangles are congruent, what can you say about their sides and angles?
All their sides are the same length, and all their angles are the same size.
If two triangles are similar, what is true about their corresponding angles?
The corresponding angles of two similar triangles are equal.
In a right triangle, if one angle is 60° and the hypotenuse is 12 cm, what is the length of the side opposite the 60° angle?
The sine of 60° is 0.866. Multiply 0.866 by 12 cm.
The side opposite the 60° angle is 10.39 cm.
Two triangles, △ABC and △DEF, are congruent. If angle A is 40°, angle B is 60°, and side AB is 5 cm, what are the measures of angle D and side DE?
Since the triangles are congruent:
Angle D = Angle A = 40°
Side DE = Side AB = 5 cm
If two circles are similar, what can you say about their radii?
The radii of two similar circles are in the same ratio or proportion.
In a right triangle, if one angle is 45° and the hypotenuse is 10 cm, what is the length of the side opposite the 45° angle?
The sine of 45° is approximately 0.707. Multiply 0.707 by 10 cm.
The side opposite the 45° angle is 7.07 cm.
Two congruent rectangles have side lengths of 4 cm and 6 cm. What are the corresponding side lengths of the second rectangle?
Since the rectangles are congruent, the corresponding side lengths of the second rectangle are also:
One side = 4 cm
Other side = 6 cm
Two similar triangles have a scale factor of 2:3. If one side of the smaller triangle is 4 cm, what is the length of the corresponding side in the larger triangle?
The corresponding side in the larger triangle is 6 cm.
In a right triangle, one angle is 40° and the hypotenuse is 15 cm. What is the length of the side opposite the 40° angle?
The sine of 40° is 0.6428. Multiply 0.6428 by 15 cm.
The side opposite the 40° angle is 9.64 cm.