Figure ABCD is transformed to obtain figure A′B′C′D′:
A coordinate grid is shown from negative 6 to 6 on both axes at increments of 1. Figure ABCD has A at ordered pair negative 4, 4, B at negative 2, 2, C at negative 2, negative 1, D at negative 4, 1. Figure A prime B prime C prime D prime has A prime at ordered pair 4, 1, B prime at 2, negative 1, C prime at 2, negative 4, D prime at 4, negative 2.
Part A: Write the sequence of transformations that changes figure ABCD to figure A′B′C′D′. Explain your answer and write the coordinates of the figure obtained after each transformation. (6 points)
Part B: Are the two figures congruent? Explain your answer. (4 points)
Part A: The sequence of transformations that changes figure ABCD to figure A′B′C′D′ is:
1. Translation: A translation of 8 units to the right and 3 units down. This moves point A from (-4, 4) to (4, 1), point B from (-2, 2) to (2, -1), point C from (-2, -1) to (2, -4), and point D from (-4, 1) to (4, -2).
2. Reflection: A reflection over the x-axis. This flips the y-coordinates of the points, resulting in point A′ at (4, -1), point B′ at (2, 1), point C′ at (2, 4), and point D′ at (4, 2).
Therefore, the sequence of transformations is a translation followed by a reflection.
Part B: No, the two figures are not congruent because congruent figures must have the same shape and size. Although the figures have the same shape, they do not have the same size since the coordinates of the vertices are different.
In the figure shown, line AB is parallel to line CD.
Part A: What is the measure of angle x? Show your work. (5 points)
Part B: Explain how you found the measure of angle x by identifying the angle relationships that you used along the transversal. (5 points)
AB and CD are parallel lines, and PQ and PR are transversals which intersect AB at P and CD at Q and R. Angle APQ is labeled as 62 degrees, angle QPR is equal to x, angle PRD is equal to 115 degrees.
Part A: To find the measure of angle x, we can use the fact that corresponding angles formed by a transversal crossing parallel lines are congruent. Therefore, angle x is congruent to angle QPR, which is labeled as x.
Since angle PRD is given as 115 degrees, we can use the sum of angles in a triangle to find angle QPR. The sum of angles in a triangle is always 180 degrees.
The sum of angle APQ (62 degrees) + angle QPR (x) + angle PRD (115 degrees) = 180 degrees.
62 + x + 115 = 180.
Combining like terms:
x + 177 = 180.
Subtracting 177 from both sides:
x = 180 - 177.
x = 3 degrees.
Therefore, the measure of angle x is 3 degrees.
Part B: To find the measure of angle x, I used the angle relationships formed by a transversal crossing parallel lines. Specifically, I used the fact that corresponding angles formed by the transversal (angle APQ and angle QPR) are congruent.
I also used the fact that the sum of angles in a triangle is always 180 degrees. By adding angle APQ, angle QPR, and angle PRD, I found the sum of angles in the triangle formed by line PR and transversal PQ and calculated the value of angle QPR, which is equal to x.
The figure shows triangle ABC and line segment PQ, which is parallel to BC:
Triangle ABC has a point P on side AB and point Q on side AC. The line PQ is parallel to the line BC.
Part A: Is triangle ABC similar to triangle APQ? Explain using what you know about triangle similarity. (5 points)
Part B: Which line segment on triangle APQ corresponds to line segment BC? Explain your answer. (3 points)
Part C: Which angle on triangle APQ corresponds to angle B? Explain your answer. (2 points)
Part A: Yes, triangle ABC is similar to triangle APQ.
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. In this case, we can see that angle A in triangle ABC is congruent to angle A in triangle APQ because they are corresponding angles formed by parallel lines BC and PQ. Similarly, angle C in triangle ABC is congruent to angle Q in triangle APQ.
To show that the sides are proportional, we can use the fact that PQ is parallel to BC. This implies that triangle APQ and triangle ABC form transversals of the parallel lines BC and PQ. Therefore, if we consider the ratios of corresponding side lengths, we have:
PQ / BC = PA / AB = AQ / AC.
Since the ratios of corresponding side lengths are equal, we can conclude that triangle ABC and triangle APQ are similar.
Part B: The line segment on triangle APQ that corresponds to line segment BC is segment PQ.
In similar triangles, corresponding sides have proportional lengths. In this case, the corresponding side to BC in triangle APQ is PQ, as they are opposite sides of parallel lines BC and PQ.
Part C: The angle on triangle APQ that corresponds to angle B is angle P.
This is because angle B in triangle ABC and angle P in triangle APQ are corresponding angles formed by parallel lines BC and PQ. In similar triangles, corresponding angles are congruent. Therefore, angle P corresponds to angle B.
The figure shows triangle PQR and line segment AB, which is parallel to QR:
Triangle PQR has a point A on side PQ and point B on side PR. The line AB is parallel to the line QR.
Part A: Is triangle PQR similar to triangle PAB? Explain using what you know about triangle similarity. (5 points)
Part B: Which line segment on triangle PAB corresponds to line segment QR? Explain your answer. (3 points)
Part C: Which angle on triangle PAB corresponds to angle Q? Explain your answer. (2 points)
Part A: Yes, triangle PQR is similar to triangle PAB.
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. In this case, we can see that angle P in triangle PQR is congruent to angle P in triangle PAB because they are corresponding angles formed by parallel lines QR and AB. Similarly, angle R in triangle PQR is congruent to angle A in triangle PAB.
To show that the sides are proportional, we can use the fact that AB is parallel to QR. This implies that triangle PAB and triangle PQR form transversals of the parallel lines AB and QR. Therefore, if we consider the ratios of corresponding side lengths, we have:
PA / PQ = AB / QR = PB / PR.
Since the ratios of corresponding side lengths are equal, we can conclude that triangle PQR and triangle PAB are similar.
Part B: The line segment on triangle PAB that corresponds to line segment QR is segment AB.
In similar triangles, corresponding sides have proportional lengths. In this case, the corresponding side to QR in triangle PAB is AB, as they are opposite sides of parallel lines QR and AB.
Part C: The angle on triangle PAB that corresponds to angle Q is angle P.
This is because angle Q in triangle PQR and angle P in triangle PAB are corresponding angles formed by parallel lines QR and AB. In similar triangles, corresponding angles are congruent. Therefore, angle P corresponds to angle Q.