C is the circumcenter of isosceles triangle ABD with vertex angle ∠ABD. Does the following proof correctly justify that triangles ABE and DBE are congruent?
It is given that triangle ABD is an isosceles triangle, so segments AB and DB are congruent by the definition of isosceles triangle.
It is given that C is the circumcenter of triangle ABD, making segment BE a median.
By the definition of perpendicular, angles AEB and DEB are 90°, so triangles ABE and DEB are right triangles.
Triangles ABE and DEB share side BE making it congruent to itself by the reflexive property.
Triangles ABE and DBE are congruent by HL.
Triangle ABD with segments BC, DC, and AC drawn from each vertex and meeting at point C inside triangle ABD, segment BC is extended past C with dashed lines so that it intersects with side AD at point E.
There is an error in line 1; segments AB and BD are given to be congruent.
There is an error in line 2; segment BE should be a perpendicular bisector.
There is an error in line 4; segment BE is not a shared side.
The proof is correct.
Question 5(Multiple Choice Worth 4 points)
(02.01 LC)
Figure EFGHK as shown below is to be transformed to figure E′F′G′H′K′ using the rule (x, y) → (x + 8, y + 5):
Figure EFGHK is drawn on a 4 quadrant coordinate grid with vertices at E 3, negative 4. F is at 5, 1. G is at 3, 5. H is at negative 4, 3. K is at negative 2, negative 3.
Which coordinates will best represent point H′?
(4, 8)
(1, 11)
(12, 8)
(9, 11)
Question 6(Multiple Choice Worth 4 points)
(02.04 MC)
What conclusion can be made for c and e?
Triangle ABC has side AB measuring 2 and a half units, side BC measuring 4 units, and angle ACB measuring e degrees. Triangle ABC is connected to triangle ACD by shared side AC. Triangle ACD has side AD measuring 2 and 8 tenths units and angle ACD measuring c degrees. Segments BC, AC and DC are marked congruent.
c > e
c < e
c ≥ e
c ≤ e
Question 7(Multiple Choice Worth 4 points)
(02.03 MC)
Which of the following would be a line of reflection that would map ABCD onto itself?
Square ABCD with A at negative 3 comma 0, B at negative 3 comma 2, C at negative 1 comma 2, and D at negative 1 comma 0.
y = 2
3x + y = 1
3x + y = −1
−3x + 3y = 9
Question 8(Multiple Choice Worth 4 points)
(02.06 MC)
Figure ABCD is a rhombus, and m∠AEB = 7x + 6. Solve for x.
Rhombus ABCD with diagonals AC and BD and point E as the point of intersection of the diagonals.
5.56
12
24.85
Not enough information
Question 9(Multiple Choice Worth 4 points)
(02.06 MC)
Look at the quadrilateral shown below:
A quadrilateral ABCD is shown with diagonals AC and BD intersecting in point O. Angle AOB is labeled as 1, angle BOC is labeled as 4, angle COD is labeled as 2, and angle AOD is labeled as 3.
Terra writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram:
Terra's proof
AO = OC because it is given that diagonals bisect each other.
BO = OD because it is given that diagonals bisect each other.
For triangles AOB and COD, angle 1 is equal to angle 2, as they are ________.
Therefore, the triangles AOB and COD are congruent by SAS postulate.
Similarly, triangles AOD and COB are congruent.
By CPCTC, angle ABD is equal to angle BDC and angle ADB is equal to angle DBC.
As the alternate interior angles are congruent, the opposite sides of quadrilateral ABCD are parallel.
Therefore, ABCD is a parallelogram.
Which is the missing phrase in Terra's proof?
alternate interior angles
corresponding angles
same-side interior angles
vertical angles
Question 10(Multiple Choice Worth 4 points)
(02.03 MC)
A land surveyor places two stakes 500 ft apart and locates the midpoint between the stakes. From the midpoint, he needs to place another stake 100 ft away that is equidistant to the two original stakes. To apply the Perpendicular Bisector Theorem, the land surveyor would need to identify a line that is
perpendicular to the line connecting the two stakes and going through the midpoint of the two stakes
parallel to the line connecting the two stakes and going through the midpoint of the two stakes
perpendicular to the line connecting the two stakes and going through one of the two original stakes
parallel to the line connecting the two stakes and going through one of the two original stakes
Question 11(Multiple Choice Worth 4 points)
(02.04 MC)
Jeremiah is working on a model bridge. He needs to create triangular components, and he plans to use toothpicks. He finds three toothpicks of lengths 4 in., 5 in., and 1 in. Will he be able to create the triangular component with these toothpicks without modifying any of the lengths?
Yes, according to the Triangle Inequality Theorem
Yes, according to the Triangle Sum Theorem
No, according to the Triangle Inequality Theorem
No, according to the Triangle Sum Theorem
Question 12(Multiple Choice Worth 4 points)
(02.02 MC)
Ben performed a transformation on trapezoid PQRS to create P′Q′R′S′, as shown in the figure below:
A four quadrant coordinate grid is drawn. Trapezoid PQRS with coordinates at P negative 6, negative 3. Q is at negative 4, negative 3. R is at negative 2, negative 5. S is at negative 7, negative 6. Trapezoid P prime Q prime R prime S prime with coordinates are drawn at P prime 3, negative 6. Q prime is at 3, negative 4. R prime is at 5, negative 2. S prime is at 6, negative 7.
What transformation did Ben perform to create P′Q′R′S′?
Rotation of 270° counterclockwise about the origin
Reflection across the line of symmetry of the figure
Reflection across the y-axis
Rotation of 90° counterclockwise about the origin
Question 13(Multiple Choice Worth 4 points)
(02.04 MC)
Charlotte is writing statements to prove that the sum of the measures of interior angles of triangle PQR is equal to 180°. Line m is parallel to line n.
Line n is parallel to line m. Triangle PQR has vertex P on line n and vertices Q and R on line m. Angle QPR is 70 degrees. Segment PQ makes 35 degree angle with line n and segment PR makes 75 degree angle with line n.
Which is a true statement she could write?
Angle PRQ measures 35°.
Angle PRQ measures 75°.
Angle PQR measures 70°.
Angle PRQ measures 70°.
Question 14(Multiple Choice Worth 4 points)
(02.03 MC)
Angle N = 40 degrees, side NP = 8, angle Q = 40 degrees, and side QS = 8. What additional information would you need to prove that ΔNOP ≅ ΔQRS by ASA?
Angle O is congruent to angle R.
Angle P is congruent to angle S.
Side NO is congruent to side QR.
Side OP is congruent to side RS.
Question 15(Multiple Choice Worth 4 points)
(02.01, 02.02 MC)
Triangle XYZ is shown on the coordinate plane.
A triangle on the coordinate plane with vertices X at 5 comma 6, Y at 10 comma 1, and Z at 2 comma 1.
If triangle XYZ is translated using the rule (x, y) → (x + 5, y − 3) and then rotated 90° clockwise to create triangle X″Y″Z″, what is the location of X″?
(−3, 10)
(−2, −15)
(−2, −7)
(3, −10)
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FDK41.12