Name That Term
What term means a statement accepted as true without proof?
Postulate
Identify the hypothesis and conclusion:
“If an angle measures 35°, then it is acute.”
Hypothesis: The angle measures 35°.
Conclusion: It is acute.
Rewrite as an if-then statement:
“All equilateral triangles have three congruent sides.”
If a triangle is equilateral, then it has three congruent sides.
For every question, you must give a specific counterexample and explain how it disproves the claim.
“Every quadrilateral is a square.”
A nonsquare rectangle has four sides but is not a square.
Maya observes that the first five buses arriving at school are yellow. She concludes that every school bus is yellow. Is her reasoning inductive or deductive?
Inductive: she generalizes from observations.
Given m∠A=m∠B, what definition allows you to conclude ∠A≅∠B?
Definition of congruent angles
What term means a statement that has been shown to be true through a chain of reasoning?
Theorem
Write the converse:
“If a figure is a square, then it is a rectangle.”
If a figure is a rectangle, then it is a square.
Write a biconditional using this definition:
“Complementary angles are two angles whose measures sum to 90°.”
Two angles are complementary if and only if their measures sum to 90°.
For every question, you must give a specific counterexample and explain how it disproves the claim.
“If two angles are supplementary, then each measures 90°.”
Answers may vary.
Example: 60° and 120° sum to 180°, but neither is 90°.
All squares have four right angles. Figure ABCD is a square. What can you conclude about figure ABCD, and which type of reasoning is used?
ABCD has four right angles; deductive reasoning.
Two lines intersect. Angles 1 and 2 are opposite, nonadjacent angles at the intersection. What theorem proves ∠1≅∠2?
Vertical Angles Theorem
You notice a pattern and make a statement you believe is true, but have not proven. What is that statement called?
Conjecture
Write the inverse:
“If two angles are vertical angles, then they are congruent.”
If two angles are not vertical angles, then they are not congruent.
Is this biconditional true? Explain:
“An angle is obtuse if and only if its measure is greater than 90°.”
No. An obtuse angle must also measure less than 180°. For example, a 180° angle is greater than 90° but is not obtuse.
For every question, you must give a specific counterexample and explain how it disproves the claim.
“If two angles are congruent, then they are vertical angles.”
Answers may vary.
Example: Two separate 40° angles are congruent but need not be formed by intersecting lines.
Find the first invalid UNO move and explain:
Blue 4 → Red 4 → Red 8 → Yellow 2
Red 8 → Yellow 2 is invalid because neither the color nor the number matches.
Complete the reasons: Two perpendicular lines form angles A and B. Therefore, both angles measure 90° because ______. Since their measures are equal, the angles are congruent because ______.
Complete the reasons: Two perpendicular lines form angles A and B. Therefore, both angles measure 90° because perpendicular lines form right angles, which measure 90°. Since their measures are equal, the angles are congruent because definition of congruent angles.
Which type of reasoning uses specific observations to reach a general conclusion? Which uses facts and general principles to reach a conclusion?
Inductive reasoning; Deductive reasoning
Write the contrapositive:
“If an angle is a right angle, then it measures 90°.”
If an angle does not measure 90°, then it is not a right angle.
For a square, write the converse. If it is true, write the biconditional, if not provide a counterexample:
“If its side length is 6 cm, then its perimeter is 24 cm.”
Converse: If its perimeter is 24 cm, then its side length is 6 cm.
Biconditional: A square has side length 6 cm iff its perimeter is 24 cm.
A new toy called “Blips” has blown up on TikTok. All Blips are orange. A student concludes that every orange object must be a Blip. Describe a situation showing that conclusion does not necessarily follow.
Answers may vary.
Example: An orange basketball that is not a Blip. All Blips could still be orange.
If an animal is a dolphin, it is a mammal. If an animal is a mammal, it breathes air. Luna is a dolphin. What follows? Also write one conditional summarizing the chain.
Luna breathes air.
Conditional: If an animal is a dolphin, then it breathes air.
Points A,B,C,D lie on a line in that order. Given AB=CD, complete the missing reasons.
| Statement | Reason |
|---|---|
| AB=CD | Given |
| AB+BC=CD+BC | ? |
| AB+BC=AC and CD+BC=BD | ? |
| AC=BD | Substitution Property of Equality |
Reason 1: Addition Property of Equality
Reason 2: Segment Addition Postulate
Name both:
(a) A sequence of statements in which each follows logically from the previous statements.
(b) A logical argument establishing that a mathematical statement is true.
(a) Logical chain;
(b) Proof
Negate this statement:
“All students in the club own a bicycle.”
At least one student in the club does not own a bicycle.
A student writes:
“Two angles are supplementary if and only if they form a linear pair.”
1. Write the two conditional statements contained in this biconditional.
2. Identify which direction is false and give a counterexample.
3. Replace either the hypothesis or conclusion with a condition that makes the biconditional true.
1. If two angles are supplementary, then they form a linear pair; If two angles form a linear pair, then they are supplementary.
2. The first is false. Answers may vary. (Example: Two nonadjacent angles measuring 60° and 120° are supplementary but do not form a linear pair.)
3. “Two angles are supplementary if and only if their measures sum to 180°.”
For every question, you must give a specific counterexample and explain how it disproves the claim.
“If two angles are complementary, then they form a right angle together as adjacent angles.”
Answers may vary.
Example: Two nonadjacent angles measuring 30° and 60° are complementary, but they do not form one right angle together.
The UNO sequence below is invalid:
Green 6 → Red 6 → Yellow 9 → Yellow 2
Find all three ways to make the entire sequence valid by replacing exactly one card with another non-Wild card. Explain why each replacement works.
Yellow 9 → Red 2: matches Red 6 by color and Yellow 2 by number.
Yellow 9 → Yellow 6: matches Red 6 by number and Yellow 2 by color.
Red 6 → Yellow 6: matches Green 6 by number and Yellow 9 by color.
Angles 1 and 2 are complementary. Angles 2 and 3 are also complementary.
Complete the missing reasons to prove \angle1\cong\angle3.
| Statement | Reason |
|---|---|
| m\angle1+m\angle2=90° and m\angle2+m\angle3=90° | Definition of complementary angles |
| m\angle1+m\angle2=m\angle2+m\angle3 | . ① . |
| m\angle1=m\angle3 | . ② . |
| \angle1\cong\angle3 | . ③ . |
① Transitive Property of Equality
② Subtraction Property of Equality
③ Definition of congruent angles