Proofs Flashcards
Given: Vertical Angles
Vertical Angles are Congruent
What should be done when given that segments bisect?
GIVEN: __ & __ bisects each other OR ___ bisects ____
1. Statement: ______ is the midpoint
1. Reason: A segment bisector meets at the midpoint
2. Statement: ______ congruent ________
2. A midpoint divides a segment into two congruent segments
How can supplementary angles be used in a proof? (4)
Two angles must have been proven congruent (X & Y)
- Statement: X & A are supplementary and Y & Z are supplementary
- Reason: Linear Pairs are Supplementary
- Statement: A = Z
- Supplements of congruent angles are congruent
What should be done when given that angles bisect?
GIVEN: ___ bisects ____
Statement: ______ = _____
Reason: An angle bisector divides an angle into two congruent angles
What must be done before proving two sides are congruent through addition and substractiction?
AC - BC = EC - DC
AB = ED
AC must be proven congruent to EC
BC must be proven congruent to DC
How do you prove right angles?
Perpendicular Lines form right angles
How do you prove that the parts of two congruent sides are congruent?
Given: AB = CD, E mp of AB and F mp of CD
(Note: DO NOT NEED TO PROVE THAT THE PARTS OF THE SAME SIDEARE CONGRUENT)
- Statement: EB = 0.5 AB & FD = 0.5 CD
- Reason: Midpoint divides a segment in one half
- Statement: 0.5 AB = 0.5 CD
- Reason: Multiplication
- Statement: EB = FD
- Reason: Substitution
How can an Isosceles Triangles be used in Proofs? (2)
- In a triangle, angles opposite congruent sides are congruent
- In a triangle, sides opposite congruent angles are congruent
How do you prove parallel lines? (2)
- If 2 lines are cut by trans. such that (alt. int./corr./alt. ext.) angles are congruent, lines are parallel
- If 2 lines are cut by trans. such that cons. int. angles are supp., lines are parallel
How do you use parallel lines in a proof? (2)
- If 2 parallel lines are cut by trans, then (alt. int./corr./alt. ext.) angles are congruent
- If 2 parallel lines are cut by trans, cons. int. angles are supplementary
How to prove two triangles are congruent?
- SAS
- ASA
- AAS
- HL
How to prove two triangles are similar in proofs?
AA similarity Theorem
2 angles in one triangle are congruent to 2 angles in another triangle
How do you utilize parts of congruent & similar triangles?
- Corresponding Parts of Congruent Triangles are Congruent
- Corresponding Parts of Similar Triangles are Similar
How do you prove that two sides of similar triangles are in proportion and the products of the sides are in proportion?
- Prove Triangles are Similar through AA Sim. Theorem
- Corresponding Sides o Similar Triangles are in Proportion
- Products of Means = Products of Extremes
What are three ways to prove an isosceles trapezoid?
- Must have at least one pair of congruent opposite sides
- Each pair of base angles are congruent
- Diagonals are congruent
How can you prove a rectangle? (2)
Prove that is a parallelogram first, then use the following two properties to prove that the parallelogram is a rectangle:
- One right angle (rect. has 4 rt. angles)
- Diagonals are congruent (rect. is also an isoc. trapezoid)
What are the five ways to prove a parallelogram?
- If both pairs of opposite sides are congruent , then the quad is a parallelogram
- If both pairs of opposite sides are parallel, then the quad is a parallelogram
- If both pairs of opposite angles are congruent, then the quad is a parallelogram
- If diagonals bisect each other, then the quad is a parallelogram (prove the mp)
- If ONE pair of opposite sides are both parallel & congruent, then the quad is a parallelogram
What is something to keep in mind with when proving properties of a quadrilateral?
You may use all the properties that the quadrilateral posess except for the properties you are trying to prove.
How can you prove a rhombus? (3)
Prove that is a parallelogram first, then use the following three properties to prove that the parallelogram is a rhombus:
1. Diagonals are perpendicular
2. Pair of consecutive sides are congruent
3. A diagonal bisects one set of angles
How do you prove a square? (4)
- Prove it is a parallelogram
- Prove it is a rectangle
- Prove it is a rhombus
- A parallelogram has one property of a rhombus & rectangle, it’s a square
How to prove a quad. to be rhombus thru perpendicular diagonals
(When they give you the rt. angle)
Statement - Diagonals (Segments name) are perpendicular
Reasoning - Perpendicular Lines form Right Angles
Statement - A quad is a rhombus
Reasoning - In kite/rhombus, diagonals are perpendicular to each other
Reason for when you add to angles it forms the whole angle?
Angle Addition Postulate
