Introduction to Congruence

Show your work for each problem.

1.
What does it mean when two shapes are congruent?
1. a)They have the same number of sides
2. b)They have the same size and shape
3. c)They have the same color
4. d)They have the same angles only
Answer: They have the same size and shape
Congruent shapes have exactly the same size and shape. One can be placed perfectly on top of the other.
2.
Which symbol is used to show that two figures are congruent?
1. a)
2. b)
3. c)
4. d)
Answer:
The congruence symbol is . We write to show the triangles are congruent. The symbol means similar, and is for numbers.
3.
Triangle ABC has sides 4 cm, 6 cm, and 8 cm. Triangle DEF is congruent to ABC. What is the length of the longest side of triangle DEF (in cm)?
Answer: 8
Since the triangles are congruent, they have the same side lengths. The longest side of ABC is 8 cm, so the longest side of DEF is also 8 cm.
4.
Two triangles have the same three angles. Are they definitely congruent?
1. a)Only if one angle is 90 degrees
2. b)Yes, same angles means congruent
3. c)No, they could be different sizes
4. d)Only if they have 3 sides
Answer: No, they could be different sizes
Same angles means the triangles are similar (same shape), but they could be different sizes. A small and large triangle can have the same angles. For congruence, side lengths must also match.
5.
Rectangle ABCD is congruent to rectangle EFGH. If rectangle ABCD has a perimeter of 24 cm, what is the perimeter of rectangle EFGH (in cm)?
Answer: 24
Congruent figures have exactly the same size and shape, which means they have the same perimeter. If ABCD has a perimeter of 24 cm, then EFGH also has a perimeter of 24 cm.
6.
A shape is rotated 90 degrees. Is the rotated shape congruent to the original?
1. a)No, the orientation changed
2. b)Only for squares
3. c)Only if it is rotated 180 degrees
4. d)Yes, rotation preserves congruence
Answer: Yes, rotation preserves congruence
Rotation is a rigid transformation that preserves size and shape. Only the position and orientation change, not the measurements. Therefore, a rotated shape is always congruent to the original.
7.
Triangle ABC has angles , , and . Triangle DEF has the same angles. Are they congruent? Explain.
Answer: no
1. Do the triangles have the same angles? yes
2. What do same angles tell us about the shape? same shape
3. Can triangles with the same angles have different sizes? yes
4. So are they definitely congruent? no
8.
. If angle A = and angle B = , what is the measure of angle F (in degrees)?
Answer: 50
Angle C = . In congruent triangles, corresponding angles are equal. A corresponds to D, B corresponds to E, and C corresponds to F. So angle F = angle C = .
9.
Which transformation does NOT preserve congruence?
1. a)Reflection
2. b)Translation
3. c)Dilation (scaling)
4. d)Rotation
Answer: Dilation (scaling)
Dilation (scaling) changes the size of a figure, so it does NOT preserve congruence. Rotation, reflection, and translation are rigid transformations that preserve both size and shape.
10.
Quadrilateral ABCD has sides 3 cm, 5 cm, 3 cm, 5 cm. Quadrilateral EFGH has sides 3 cm, 5 cm, 3 cm, 5 cm. Are they congruent?
Answer: not necessarily
1. Do both quadrilaterals have the same side lengths? yes
2. Can a quadrilateral with these sides be a rectangle? yes
3. Can a quadrilateral with these sides be a parallelogram that is not a rectangle? yes
4. If one is a rectangle and one is a tilted parallelogram, are they congruent? no
11.
. If side AB = 12 cm, what is the length of side XY (in cm)?
Answer: 12
In the notation , the order of letters shows which parts correspond. A corresponds to X, B corresponds to Y, C corresponds to Z. So side AB corresponds to side XY, and XY = 12 cm.
12.
Triangle ABC has sides 5 cm, 12 cm, and 13 cm. Triangle DEF has sides 5 cm, 12 cm, and 13 cm. Are the triangles congruent? Explain why.
Answer: yes
1. List the three sides of triangle ABC 5, 12, 13
2. List the three sides of triangle DEF 5, 12, 13
3. Do all three corresponding sides match? yes
4. For triangles, if all three sides are equal, is this enough for congruence? yes
13.
A shape is reflected, then rotated 180 degrees. Is the final shape congruent to the original?
1. a)Only if the shape is symmetric
2. b)No, two transformations cancel out
3. c)Yes, both are rigid transformations
4. d)No, reflections change congruence
Answer: Yes, both are rigid transformations
Yes! Both reflection and rotation are rigid transformations that preserve size and shape. Any sequence of rigid transformations produces a congruent figure. The final shape is congruent to the original.
14.
. The area of is 24 square cm. What is the area of (in square cm)?
Answer: 24
Congruent triangles have exactly the same size and shape, which means they have the same area. Since has an area of 24 square cm, also has an area of 24 square cm.
15.
Square ABCD has side length 5 cm. Rectangle EFGH has length 5 cm and width 5 cm. Are they congruent?
Answer: yes
1. What are the four sides of square ABCD? 5, 5, 5, 5
2. What are the four sides of rectangle EFGH? 5, 5, 5, 5
3. What are the angles in both shapes? 90
4. Are all sides and all angles equal in both shapes? yes
16.
Two circles have the same radius of 4 cm. Are they congruent?
1. a)Only if they are in the same position
2. b)Only if they have the same center
3. c)Yes, same radius means same size and shape
4. d)No, circles cannot be congruent
Answer: Yes, same radius means same size and shape
Yes, the circles are congruent! A circle's size and shape are completely determined by its radius. Two circles with the same radius have exactly the same size (area and circumference) and the same shape. Position doesn't affect congruence.
17.
Equilateral triangle ABC is congruent to equilateral triangle DEF. If the perimeter of ABC is 27 cm, what is the side length of DEF (in cm)?
Answer: 9
An equilateral triangle has three equal sides. If the perimeter is 27 cm, each side = cm. Since the triangles are congruent, DEF also has side length 9 cm.