Angle Relationships
Two angles that add up to 90 degrees are called what?
Complementary Angles
Two triangles share a side, and you are given two pairs of congruent angles and one pair of non-included congruent sides. Which triangle congruence theorem applies?
AAS, because the side is not included between the angles; ASA would require the congruent side to be between the two angles.
What is the formula for an equation of a line?
y=mx+b
What is the transformation that flips a figure over a line.
A reflection
The sum of the interior angles of any triangle.
What is 180°?
Two angles form a linear pair. One angle measures 47°. What is the measure of its supplement?
133°
In triangle ABC and triangle DEF, you are given:
AB ≅ DE
AC ≅ DF
∠A ≅ ∠D
Can you use SAS to prove the triangles congruent? Why or why not?
No, because the angle is not included between the two sides.
What is the formula used to calculate the slope of a line.
m = (y2-y1)/(x2-x1)
When a figure is reflected across a line, what happens to the distance between any point and the line of reflection?
The point and its image are the same distance from the line of reflection
If two angles are supplementary to the same angle, what can you conclude about those two angles?
The angles are congruent
Angle A and Angle B are vertical angles. If Angle A is represented by the expression 3x + 15 and Angle B is 6x – 30, what is the value of x, and what’s the measure of each angle? Justify.
x = 15, and both angles measure 60° because vertical angles are congruent
A student says:
“△XYZ ≅ △PQR because all three pairs of angles are congruent.”
Is this reasoning valid? Why or why not? What misconception might the student have?
No, the reasoning is invalid. AAA (Angle-Angle-Angle) does not prove triangle congruence, only similarity. The student confused angle congruence with triangle congruence, which requires side information.
riangle A has vertices at (1, 1), (1, 5), and (4, 1). Triangle B has vertices at (–1, 1), (–1, 5), and (–4, 1).
Without using formulas, explain how you know the two triangles are congruent.
The triangles are mirror images (reflections) across the y-axis, which is a rigid motion and preserves congruence.
Explain why any rigid motion (translation, reflection, or rotation) produces an image congruent to the original figure.
Rigid motions preserve distances and angle measures, the image and pre-image have the same size and shape, making them congruent
What does CPCTC stand for and how is it used in a proof?
Corresponding Parts of Congruent Triangles are Congruent; it is used to justify that parts (angles or sides) of congruent triangles are equal?
In triangle XYZ, ∠X = 2x + 10, ∠Y = 3x – 5, and ∠Z = x + 25. Find the measure of each angle in the triangle.
∠X = 50°, ∠Y = 70°, ∠Z = 60°, because the angles in a triangle add up to 180°
You are given triangles △PQR and △STU with the following information from a diagram:
PQ ≅ ST
∠P ≅ ∠S
QR ≅ TU
Can you prove the triangles are congruent? If so, which postulate applies? If not, explain what information is missing or why it cannot be used.
No, you cannot prove triangle congruence with this information. The angle given (∠P ≅ ∠S) is not included between the two given sides, so SAS does not apply. Without knowing if ∠Q ≅ ∠T or PR ≅ SU, there isn’t enough information to justify congruence using any triangle congruence postulate.
A figure is flipped over the y-axis on the coordinate plane. Describe how the coordinates of any point (x, y) change after this reflection.
The x-coordinate changes sign, so (x, y) becomes
(–x, y)
A figure is rotated 270° counterclockwise about a point. Describe how this rotation compares to a 90° clockwise rotation and explain why.
They produce the same result because rotating 270° CCW is equivalent to rotating 90° CW around the same point
In an isosceles triangle, what theorem explains the relationship between the angles opposite the equal sides?
The Base Angles Theorem (the angles opposite the equal sides are congruent)
In quadrilateral ABCD, three angles measure 92°, 88°, and 76°. The fourth angle is split into two adjacent angles in a linear pair. One part is twice the size of the other. What is the measure of each of those two angles?
34.6 and 69.3
In the diagram, triangles △ABC and △CBD share side BC.
You are told:
AB ≅ DB
∠A ≅ ∠D
Point B is the midpoint of segment AC
Can you prove △ABC ≅ △CBD? State the congruence postulate and justify your reasoning clearly.
Yes, △ABC ≅ △CBD by SAS.
AB ≅ DB (given)
∠A ≅ ∠D (given)
B is the midpoint of AC, so BC ≅ BC (reflexive or shared side).
These three pairs form Side-Angle-Side, where the included angle is congruent.
If two lines have the same slope, they are
Parallel
If you rotate a figure 90° clockwise about a point and then rotate it 180° counterclockwise about the same point, what single rotation is equivalent to these two combined rotations?
A 90° counterclockwise rotation?
(Because 90° CW + 180° CCW = 90° CCW)
What does the Triangle Inequality Theorem state about the lengths of the sides of a triangle?
The sum of the lengths of any two sides of a triangle is greater than the length of the third side?