A triangle RPQ is shown. S is a point on side PQ, and T is a point on side PR. Points S and T are joined using a straight line. The length of PS is equal to 28, the length of SQ is equal to 12, the length of PT is equal to x, and the length of TR is equal to 15.
Patricia is writing statements as shown below to prove that if segment ST is parallel to segment RQ, then x = 35:
Statement Reason
1. Segment ST is parallel to segment QR. Given
2. Angle QRT is congruent to angle STP. Corresponding angles formed by parallel lines and their transversal are congruent.
3. Angle SPT is congruent to angle QPR. Reflexive property of angles
4. Triangle SPT is similar to triangle QPR. Angle-Angle Similarity Postulate
5. ? Corresponding sides of similar triangles are in proportion.
Which equation can she use as statement 5?
x:15 = 28:40
28:12 = x:(x + 15 )
x + 15 = 28 + 12
28:40 = x:(x + 15)
Question 2
(Multiple Choice Worth 1 Points)
(02.01, 02.02 MC)
Rectangle ABCD is translated (x + 2, y − 3) and then rotated 180° about the origin. Complete the table to show the locations of A″, B″, C″, and D″ after both transformations.
Rectangle ABCD is shown. A is at negative 5, 1. B is at negative 5, 3. C is at negative 1, 3. D is at negative 1, 1.
Triangle JKL and triangle GHI are shown in the figure.
Triangle G H I and triangle J K L are shown. Angle L and angle I are congruent. Angle J and G are congruent to each other. Angle K and angle H are right angles.
If m∠J = 44°, what is m∠I?
2°
44°
46°
90°
Question 4
(Multiple Choice Worth 1 Points)
(02.06 MC)
The figure below shows a parallelogram ABCD. Side AB is parallel to side DC, and side AD is parallel to side BC:
A quadrilateral ABCD is shown with the two pairs of opposite sides AD and BC and AB and DC marked parallel. The diagonals are labeled BD and AC.
A student wrote the following sentences to prove that parallelogram ABCD has two pairs of opposite sides equal:
For triangles ABD and CDB, alternate interior angle ABD is congruent to angle CDB because AB and DC are parallel lines. Similarly, alternate interior angle ADB is equal to angle CBD because AD and BC are parallel lines. DB is equal to DB by the reflexive property. Therefore, triangles ABD and CDB are congruent by the SAS postulate. Therefore, AB is congruent to DC and AD is congruent to BC by CPCTC.
Which statement best describes a flaw in the student's proof?
Angle ABD is congruent to angle CBD because they are vertical angles, not alternate interior angles.
Angle ABD is congruent to angle CBD because they are corresponding angles, not alternate interior angles.
Triangles ABD and CDB are congruent by the SSS postulate instead of the SAS postulate.
Triangles ABD and CDB are congruent by the ASA postulate instead of the SAS postulate.
Question 5
(Multiple Choice Worth 1 Points)
(01.02 MC)
Dan uses a compass to draw an arc from Q as shown. He wants to construct a line segment through R that makes the same angle with line segment QR as line segment PQ, as shown below:
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. The ends of a compass are placed at Q and S.
Which figure shows the next step to construct a congruent angle at R?
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. Another arc drawn from point R is a reflection of the first arc but of the same measure and intersects QR. The compass is placed at point R to draw this arc.
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. Another arc drawn from point R is a reflection of the first arc but of the same measure and intersects QR. A line segment RX is drawn that makes the same angle with QR as QP makes with QR. A straight edge is placed along the length of RX.
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. Another arc drawn from point R is a reflection of the first arc but of the same measure and intersects QR. A straight edge placed at the intersection of the arcs near R is also shown.
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. A compass is shown drawing an arc between PQ and QR from point S.
Question 6
(Multiple Choice Worth 1 Points)
(01.07 MC)
Chelsea drew two parallel lines, KL and MN, intersected by a transversal PQ, as shown below:
Two parallel lines, KL and MN, with PQ as a transversal intersecting KL at point R and MN at point S. Angle KRP is shown congruent to angle MSP.
Which fact would help Chelsea prove that the measure of angle KRP is equal to the measure of angle MSR?
When angles KRS and MSR are equal to angle RSN, the angles KRP and MSR are congruent.
When angles KRP and KRS are equal to angle LRS, the angles KRP and MSR are congruent.
When angle LRS is equal to angle KRP and angles LRS and RSN are complementary, angle KRP and angle MSR are congruent.
When angle LRS is equal to both the angles KRP and MSR, angle KRP and angle MSR are congruent to each other.
Question 7
(Multiple Choice Worth 1 Points)
(01.06 MC)
Ray IL bisects angle HIJ. If m∠HIL = (6x − 7)° and m∠JIL = (5x + 4)°, what is m∠JIL?
118°
59°
44°
11°
Question 8
(Multiple Choice Worth 1 Points)
(02.01 MC)
A triangle has vertices at A (−2, −2), B (−1, 1), and C (3, 2). Which of the following transformations produces an image with vertices A′ (2, −2), B′ (1, 1), and C′ (−3, 2)?
(x, y) → (x, −y)
(x, y) → (−y, x)
(x, y) → (−x, y)
(x, y) → (y, −x)
Question 9
(Multiple Choice Worth 1 Points)
(02.03 MC)
John translated parallelogram ABCD using the rule (x, y) → (x + 3, y − 2). If angle A is 110° and angle B is 70°, what is the degree measurement of angle A′?
70°
110°
40°
180°
Question 10
(Multiple Choice Worth 1 Points)
(03.02 MC)
Decide whether the triangles are similar. If so, determine the appropriate expression to solve for x.
Triangles ABC and EDF; triangle ABC has angle A measuring 53 degrees, angle C measuring 62 degrees, side AC labeled as y, side AB labeled as w, and side BC labeled as x; triangle EDF has angle D measuring 61 degrees, angle F measuring 53 degrees, side DE labeled z, side EF labeled u, and side DF labeled r.
The triangles are not similar; no expression for x can be found.
ΔABC ~ ΔDEF; x equals r times w over u
ΔABC ~ ΔEFD; x equals r times w over u
ΔABC ~ ΔEFD; x equals r times w over z
Question 11
(Multiple Choice Worth 1 Points)
(02.05 MC)
Triangle ABC is a right triangle. Point D is the midpoint of side AB, and point E is the midpoint of side AC. The measure of angle ADE is 68°.
Triangle ABC with segment DE. Angle ADE measures 68 degrees.
The following flowchart with missing statements and reasons proves that the measure of angle ECB is 22°:
Statement, Measure of angle ADE is 68 degrees, Reason, Given, and Statement, Measure of angle DAE is 90 degrees, Reason, Definition of right angle, leading to Statement 3 and Reason 2, which further leads to Statement, Measure of angle ECB is 22 degrees, Reason, Substitution Property. Statement, Segment DE joins the midpoints of segment AB and AC, Reason, Given, leading to Statement, Segment DE is parallel to segment BC, Reason, Midsegment theorem, which leads to Angle ECB is congruent to angle AED, Reason 1, which further leads to Statement, Measure of angle ECB is 22 degrees, Reason, Substitution Property.
Which statement and reason can be used to fill in the numbered blank spaces?
Corresponding angles are congruent
Triangle Sum Theorem
Measure of angle AED is 22°
Corresponding angles are congruent
Base Angle Theorem
Measure of angle AED is 68°
Alternate interior angles are congruent
Triangle Sum Theorem
Measure of angle AED is 22°
Alternate interior angles are congruent
Triangle Angle Sum Theorem
Measure of angle AED is 68°
Question 12
(Multiple Choice Worth 1 Points)
(02.01, 02.02 MC)
Which sequence of transformations will map figure Q onto figure Q′?
Two congruent quadrilaterals are shown on a coordinate plane; quadrilateral Q with coordinates negative 9 comma 2, negative 6 comma 4, negative 4 comma 4, and negative 2 comma 2; quadrilateral Q prime with coordinates 2 comma 2, 4 comma 4, 6 comma 4, and 9 comma 2.
Translation of (x, y + 2), reflection over x = 1, and 180° rotation about the origin
Translation of (x, y − 2), reflection over x = 1, and 180° rotation about the origin
Translation of (x, y − 2), reflection over y = 1, and 180° rotation about the origin
Translation of (x, y + 2), reflection over y = 1, and 180° rotation about the origin
Question 13
(Multiple Choice Worth 1 Points)
(01.05 MC)
What is the converse of the following statement?
"If the sum of two angles is 180°, then the angles are supplementary."
If the sum of two angles is not 180°, then the angles are not supplementary.
If the angles are supplementary, then the sum of the two angles is 180°.
If the angles are not supplementary, then the sum of the two angles is not 180°.
The sum of two angles is 180° if and only if the angles are supplementary.
Question 14
(Multiple Choice Worth 1 Points)
(03.01 MC)
Triangle ABC is shown on the coordinate plane.
A triangle on the coordinate plane with vertices A at negative 3 comma 3, B at negative 1 comma 2, and C at 0 comma 4.
If ΔABC is reflected over the y-axis and then dilated by a scale factor of 3 about the origin, where are the vertices of ΔA″B″C″ located?
(3,3), (1, 2), and (0,4)
(6, 6), (2, 4), and (0, 8)
(9, 9), (3, 6), and (0, 12)
(−9, −9), (−3, −6), and (0, −12)
Question 15
(Multiple Choice Worth 1 Points)
(02.03 MC)
Triangle MOP has been rotated to create triangle M′O′P′. MP = M′P′ = 2 units, MO = M′O′ = 2 units, and OP = O′P′ = 2.82 units. Which postulate below would prove the two triangles are congruent?
SSS
SAS
ASA
AAS
1 answer
Let's go through the questions one by one.
Question 1
Which equation can she use as statement 5?
The correct answer is:
28:12 = x:(x + 15 )
Explanation: This equation represents the proportional sides of the similar triangles (PS/SQ = PT/TR).
Question 2
Complete the table to show the locations of A″, B″, C″, and D″ after both transformations.
A (−5, 1) → A″ (−2, −3)
B (−5, 3) → B″ (−2, 1)
C (−1, 3) → C″ (0, 1)
D (−1, 1) → D″ (0, −3)
Thus, the correct transformed coordinates are:
A″ (−2, −3), B″ (−2, 1), C″ (0, 1), D″ (0, −3).
Question 3
What is m∠I?
If m∠J = 44°, then by the properties of similar triangles and right angles (as L and I are congruent, and J and G are congruent), m∠I = 44°.
So, the answer is 44°.
Question 4
Which statement best describes a flaw in the student's proof?
Angle ABD is congruent to angle CBD because they are vertical angles, not alternate interior angles.
Question 5
Which figure shows the next step to construct a congruent angle at R?
The correct answer is:
A line segment QR is drawn. PQ is a line segment that makes an obtuse angle of about 120 degrees with QR. An arc, indicating the measure of angle PQR, intersects PQ at S and QR at T. Another arc drawn from point R is a reflection of the first arc but of the same measure and intersects QR. A line segment RX is drawn that makes the same angle with QR as QP makes with QR. A straight edge is placed along the length of RX.
Question 6
Which fact would help Chelsea prove that the measure of angle KRP is equal to the measure of angle MSR?
When angle LRS is equal to angle KRP and angles LRS and RSN are complementary, angle KRP and angle MSR are congruent.
Question 7
What is m∠JIL?
We have m∠HIL = m∠JIL, which is given as:
\( (6x - 7) = (5x + 4) \)
Solving for x gives \( x = 11 \), thus \( m∠JIL = 5(11) + 4 = 59° \).
So, the answer is 59°.
Question 8
Which transformation produces the required image?
The transformation:
(x, y) → (−x, y) yields the image desired.
Question 9
What is the degree measurement of angle A′?
Angle A remains unchanged by translation, so the degree measure is still 110°.
Question 10
Are triangles similar?
Given the angles,
ΔABC ~ ΔEFD; x equals r times w over u
Question 11
Which statement and reason can be used to fill in the numbered blank spaces?
Corresponding angles are congruent.
Question 12
Which sequence of transformations will map figure Q onto figure Q′?
The correct sequence is:
Translation of (x, y − 2), reflection over y = 1, and 180° rotation about the origin.
Question 13
What is the converse of the following statement?
If the angles are supplementary, then the sum of the two angles is 180°.
Question 14
Where are the vertices of ΔA″B″C″ located?
After reflection over the y-axis and dilation by a factor of 3, the vertices will be:
(6, 6), (2, 4), and (0, 8).
Question 15
Which postulate would prove the two triangles are congruent?
The correct postulate is:
SSS (since all sides are equal).
Let me know if there's anything else you need help with!
