Geometry Honors – Proofs
1. If D is in the interior of ABC, then
m ABD + m DBC = m ABC.
2. If M is between X and Y, then XM + MY = XY.
Postulates
• Remember….
• 1. Ruler Postulate
• 2. Segment Addition Postulate
• 3. Protractor Postulate
• 4. Angle Addition Postulate
Postulates
• Postulate 5
– Through any two points there exists exactly one line.
• Postulate 6
– A line contains at least two points.
• Postulate 7
– If two lines intersect, then their intersection is exactly
one point.
• Postulate 8
– Through any three noncollinear points there exists
exactly one plane.
• Postulate 9
– A plane contains at least three noncollinear points.
Postulates
• Postulate 10
– If two points lie in a plane, then the line
containing them lies in the plane.
• Postulate 11
– If two planes intersect, then their intersection is a
line.
Sketching the Given
Sketch a diagram showing AB intersecting CD at point E, so that AB  CD
Redraw the diagram if the given information also states that
AE @ EB
Interpret
Which of the following statements cannot be
assumed from the diagram?
All points are coplanar.
FG  CD or mCEF = 90°. C, E, and D are collinear.
CEF and FED are a linear pair.
CEF @ FED
Reason Using Properties from Algebra
• Remember…..
• Addition Property?
– If a = b, then a + c = b + c
• Subtraction Property?
– If a = b, then a – c = b – c
• Multiplication Property?
– If a = b, then ac = bc
• Division Property?
– If a = b and c ≠ 0 then a/c = b/c
• Substitution Property?
– If a = b, then a can be substituted or b in any equation or expression.
Write reasons for each step
Solve 3x + 8 = -4x - 34. Write a reason for each step.
Equation
3x + 8 = -4x - 34
3x + 8 + 4x = -4x – 34 + 4x
7x + 8 = -34
7x – 8 = -34 - 8
7x = -42
7𝑥 −42
=
7
7
x = -6
Explanation
Reason
Write original
equation.
Given
Add 4x to each
side.
Addition Property
of Equality
Combine like
terms.
Simplify.
Subtract 8 from each
side.
Combine like terms.
Divide each side by 7.
Combine like terms.
Subtraction
Property of Equality
Simplify.
Division Property
of Equality
Simplify.
Geometric Properties
• Reflexive Property of Equality
– Real Numbers
– Segment Length
– Angle Measure
For any real number a, a = a
For any segment AB, AB = AB
For any angle A, 𝒎∠𝑨 = 𝒎∠𝑨
• Symmetric Property of Equality
– Real Number
– Segment Length
– Angle Measure
𝒎∠𝑨
For any real numbers a and b, if a = b, then b = a
For any segments AB and CD, if AB = CD, then CD = AB
For any angles A and B, if 𝒎∠𝑨 = 𝒎∠𝑩, 𝒕𝒉𝒆𝒏 𝒎∠𝑩 =
• Transitive Property of Equality
– Real Number
For any real numbers a, b and c, if a = b and b = c,
then a = c
– Segment Length For any segments AB, CD and EF, if AB = CD and CD = EF,
then AB = EF
– Angle Measure For any angles A, B, and C, if 𝒎∠𝑨 =
𝒎∠𝑩 𝒂𝒏𝒅 𝒎∠𝑩 =
𝒎∠𝑪, 𝒕𝒉𝒆𝒏 𝒎∠𝑨 = 𝒎∠𝑪
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
WY = XZ
Use given information
Given
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
WY = XZ
Use given information
Given
WY = WX + XY
Add lengths of adjacent
segments
Segment Addition
Postulate
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
WY = XZ
Use given information
Given
WY = WX + XY
Add lengths of adjacent
segments
Segment Addition
Postulate
XZ = XY + YZ
Add lengths of adjacent
segments
Segment Addition
Postulate
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
WY = XZ
Use given information
Given
WY = WX + XY
Add lengths of adjacent
segments
Segment Addition
Postulate
XZ = XY + YZ
Add lengths of adjacent
segments
Segment Addition
Postulate
WX + XY = XY + YZ
Substitute WX + XY for WY
and XY + YZ for YZ.
Substitution Property of
Equality
In the diagram, WY = XZ. Show that WX = YZ.
Equation
Explanation
Reason
WY = XZ
Use given information
Given
WY = WX + XY
Add lengths of adjacent
segments
Segment Addition
Postulate
XZ = XY + YZ
Add lengths of adjacent
segments
Segment Addition
Postulate
WX + XY = XY + YZ
Substitute WX + XY for WY
and XY + YZ for YZ.
Substitution Property of
Equality
WX = YZ
Subtract XY from each side. Subtraction Property of
Equality