Unit 1
What type of reasoning uses specific examples to make a general prediction: inductive or deductive reasoning?
Inductive reasoning
how many degrees is a right angle?
90°
What is a translation in geometry?
when a figure moves without rotating or flipping it
What does “SSS” stand for in triangle congruence?
SSS = Side–Side–Side
A polygon with four sides.
What is a quadrilateral?
You notice the sequence 3, 6, 9, 12… is increasing by ?Using inductive reasoning, what is the next number in the pattern?
calculate = 12 + 3 = 15
Answers:
increases by 3
=15
Segment AB is divided by point C. AC = 5 cm, CB = 7 cm. What is the length of AB?
calculate: AB = 5 + 7 = 12 cm
Answer: AB = 12 cm
riangle ABC is reflected across the y-axis. Point A has coordinates (3, 4). What are the coordinates of the reflected point A’?
Reflection across the y-axis: x → −x, y → y
A = (3, 4) → A’ = (−3, 4)
Answer: A’ = (−3, 4)
Two triangles each have a 45° angle, a 60° angle, and a side of length 12 that is NOT between the angles.
Which congruence theorem proves they are congruent?
AAS
A rhombus has all four sides equal. If one side is 11 cm long, what is the perimeter of the rhombus?
A rhombus has 4 equal sides.
perimeter = 4 × side length
calculate: 4 x 11 = 44
(Answer: 44)
A student claims: “All multiples of 5 end in the digit 5.”
Give a counterexample that proves this statement is false.
Answer: 10 is a multiple of 5 but ends in 0, not 5
∠XYZ is made of two adjacent angles: ∠XYA = 35° and ∠AYZ = 50°. Find ∠XYZ.
calculate: ∠XYZ = 35° + 50° = 85°
Answer: ∠XYZ = 85°
Triangle XYZ has vertex X at (2, 5). It is translated 4 units right and 3 units down. What are the coordinates of the new vertex X’?
Translation rule: (x, y) → (x + h, y + k)
Right = +4, Down = −3
Answer: X = (2, 5) → X’ = (2 + 4, 5 − 3) = (6, 2)
Triangle ABC has side lengths AB = 9 cm, BC = 7 cm, and AC = 6 cm.
Triangle DEF has side lengths DE = 9 cm, EF = 7 cm, and DF = 6 cm.
Are the triangles congruent? Explain using a congruence theorem.
they are congruent triangles.
by
SSS (Side–Side–Side).
In a parallelogram one interior angle measures 62∘
What are the measures of the other three angles?
Given one angle 62∘
Then the opposite angle =62
An adjacent angle = 180∘−62 Calculate:180−62=118.
(the four angles are: 62∘, 118∘, 62∘, 118∘)
Angles A and B form a linear pair. Using deductive reasoning, what can you conclude about their measures?
Answer:
∠A + ∠B = 180°
Two lines intersect. One of the angles measures 72°. Find the measures of all other angles.
Vertical angles are opposite angles formed by intersecting lines and are congruent.
The angles adjacent to each 72° angle are supplementary (sum to 180°).
Opposite angle = 72°
Adjacent angles = 180 − 72 = 108°
Answer = 72°, 108°, 72°, 108°
Point P is at (3, 1). Rotate P 90° clockwise around the origin. What are the new coordinates P’?
90° clockwise rotation (x, y) → (y, −x)
P = (3, 1) → P’ = (1, −3)
Answer: P’ = (1, −3)
Triangle ABC has sides AB = 8 cm and AC = 6 cm. The included angle ∠A = 50°. Triangle DEF has sides DE = 8 cm, DF = 6 cm, and ∠D = 50° between them. Are the triangles congruent? Explain using a congruence theorem.
△ABC ≅ △DEF
by
SAS
An isosceles trapezoid has bases of lengths 12 m and 8 m, and its height is 5 m. Find the area.
calculate: 12+8/2 = 20/2 = 10
calculate x the Hight. 10x5=50
(Area = 50 square meters)
Angle M and Angle N are vertical angles.
m∠M = 7x − 12∘
m∠N = 4x + 18∘
Use deductive reasoning to decide what equation must be true, then solve for x
calculate:
7x−12=4x+18
3x−12=18
3x=30
Answer: x = 10
Two parallel lines are cut by a transversal. One of the angles measures 65°.
(a) Find the measure of its corresponding angle.
(b) Find the measure of its alternate interior angle.
Corresponding angles = equal → 65°
Alternate interior angles = equal → 65°
Identify the angle relationships using the transversal rules.
Apply equality for corresponding and alternate interior angles.
Answers:
(a) Corresponding angle = 65°
(b) Alternate interior angle = 65°
Triangle ABC has a vertex A at (2, 3). First, reflect A across the x-axis. Then translate the reflected point 5 units left and 2 units up. What are the final coordinates of A’’?
1. Reflection across x-axis: (x, y) → (x, −y)
A = (2, 3) → A’ = (2, −3)
2. Translation 5 left, 2 up: (x, y) → (x − 5, y + 2)
A’ = (2, −3) → A’’ = (2 − 5, −3 + 2) = (−3, −1)
Answer: (−3, −1)
In △ABC and △DEF, you are given:
AB = DE = 10 cm
∠A = ∠D = 40°
∠B = ∠E = 75°
Prove the triangles are congruent.
Which congruence rule applies, and why?
1. Two angles match:
∠A = ∠D = 40°
∠B = ∠E = 75°
2. AB = DE = 10 cm.
Answer: △ABC ≅ △DEF are ASA.
A trapezoid has bases of lengths 14 m and 6 m. Its height is 5 m. What is the area of the trapezoid?
Calculate: A= (14=6)/2 x 5
14+6=20
20/2=10
10x5=50
(Area= 50m²)