🔺 TRIANGLES
What must be true for three lengths to form a triangle?
The sum of the two shorter sides must be greater than the longest side.
What are two properties that all parallelograms share? (multiple answers)
-The opposite sides are parallel and equal
-The opposite angles are equal
-The consecutive/adjacent angles are supplementary
-If any one of the angles is a right angle, then all the other angles will a right angle
-The two diagonals bisect each other
-Each diagonal bisects the parallelogram into two congruent triangles
What is the name of a line that intersects a circle at exactly one point?
Tangent
True or False: Vertical angles are always congruent.
True
What type of transformation maintains both angle measure and side length?
Rigid transformation (reflection, rotation, or translation)
Two sides of a triangle measure 9 cm and 14 cm. What are the possible lengths for the third side?
Between 5 cm and 23 cm.
Which type(s) of quadrilaterals have perpendicular diagonals?
Rhombus, square, and kite.
Write the equation of a circle with center (–2, 3) and radius 4.
(x+2)2+(y–3)2=16
If angle A and angle B are supplementary and angle A is 3x + 10 and angle B is 5x – 30, find x and the measure of each angle.
x = 25; angle A = 85°, angle B = 95°
Triangle ABC is translated by the rule (x, y) → (x – 3, y + 5). What are the new coordinates of point C(2, –4)?
C′ = (–1, 1)
Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, AC = 10 and DE = 9, what is the length of EF?
Use the scale factor: AB:DE = 6:9 = 2:3 → EF = 8 × 3/2 = 12.
Prove that a quadrilateral with diagonals that bisect each other must be a parallelogram.
Use midpoint formula to show diagonals intersect at same point.
Find the area of a sector with central angle 90° in a circle with radius 6.
Sector area = 9π
Given that two lines are cut by a transversal and form alternate interior angles measuring (2x + 15) and (3x – 5), find x.
x = 20
A triangle is reflected across the x-axis and then rotated 90° counterclockwise about the origin. What happens to point (4, –2)?
Reflect: (4, 2), then rotate: (–2, 4)
Triangle XYZ has angles of 40°, 60°, and 80°. Describe the triangle by its sides and angles.
Scalene and acute.
A rectangle has a diagonal measuring 13 units and one side measuring 5 units. Find the length of the other side.
Use Pythagorean Theorem: x=12
A circle has a diameter of 20. Find the length of an arc intercepted by a central angle of 72°.
Arc length = 4π
In triangle DEF, angle D = 2x°, angle E = (x + 10)°, and angle F = (3x – 20)°. Find the measure of each angle.
∠D =63.33°
∠E =41.67°
∠F =75°
What is the scale factor of a dilation that maps a segment of length 5 units to a segment of length 12.5 units?
Scale factor; K = 2.5
Triangle ABC has coordinates A(–3, 1), B(2, 4), and C(1, –2). What is the best classification for the given shape, explain
Isosceles triangle; side lengths are equal.
Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(3, 3), and D(–1, 3). How would you prove if it is a trapezoid or a parallelogram.
Show only one pair of opposite sides are parallel, and other pair is not.
An inscribed angle in a circle intercepts a 120° arc. What is the measure of the inscribed angle?
60°
Angles A, B, and C form a triangle. Angle A is twice angle B, and angle C is 30° more than angle B. Find all three angles.
A = 75°, B = 37.5°, C = 67.5°
A figure is rotated 180° about the origin and then reflected across the line y = x. Describe the overall transformation rule for a point (x, y).
(x, y) → (–y, –x)