LEVEL 1: BASIC PROOF BUILDING BLOCKS
If You Know... | You Can Conclude... | Reason |
AB is shared by two triangles | AB ≅ AB | Reflexive Property |
M is midpoint of AB | AM ≅ MB | Definition of Midpoint |
BD bisects ∠ABC | ∠ABD ≅ ∠DBC | Definition of Angle Bisector |
Lines intersect | Vertical angles are congruent | Vertical Angles Theorem |
Angles form a straight line | Sum = 180° | Linear Pair Theorem |
Two angles form a full circle | Sum = 360° | Angle Addition |
A=B and B=C | A=C | Transitive Property |
A=B | B=A | Symmetric Property |
A=B and equation contains A | Replace A with B | Substitution |
LEVEL 2: PARALLEL LINES & TRANSVERSALS
If You Know... | You Can Conclude... |
Lines are parallel | Corresponding angles congruent |
Lines are parallel | Alternate interior angles congruent |
Lines are parallel | Alternate exterior angles congruent |
Lines are parallel | Same-side interior angles supplementary |
Converse:
If You Prove... | You Can Conclude... |
Corresponding angles congruent | Lines are parallel |
Alternate interior angles congruent | Lines are parallel |
Alternate exterior angles congruent | Lines are parallel |
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LEVEL 3: TRIANGLE THEOREMS
Triangle Angle Sum (A+B+C=180)
Exterior Angle Theorem
If You Know... | You Can Conclude... |
Exterior angle of triangle | Equals sum of remote interior angles |
Isosceles Triangle Theorems
Forward
If You Prove... | You Can Conclude... |
Two sides congruent | Base angles congruent |
Converse
If You Prove... | You Can Conclude... |
Two angles congruent | Opposite sides congruent |
LEVEL 4: TRIANGLE CONGRUENCE
SSS
Three sides congruent
⇒ Triangles congruent
SAS
Two sides and included angle congruent
⇒ Triangles congruent
ASA
Two angles and included side congruent
⇒ Triangles congruent
AAS
Two angles and non-included side congruent
⇒ Triangles congruent
HL
Right triangles only.
Hypotenuse and leg congruent
⇒ Triangles congruent
Once we know triangles are congruent:
If You Prove... | You Can Conclude... |
Triangle congruence | Corresponding sides congruent |
Triangle congruence | Corresponding angles congruent |
LEVEL 5: TRIANGLE SIMILARITY
AA Similarity
Two angles congruent
⇒ Triangles similar
SAS Similarity
Two sides are proportional (scale factor relationship) + one angle congruent
⇒ Triangles similar
If You Prove... | You Can Conclude... |
Triangles Similar | Corresponding angles congruent |
Triangles Similar | Corresponding sides proportional |
Triangles Similar | Scale factor relationships exist amongst all corresponding sides |
LEVEL 6: COORDINATE GEOMETRY TOOLBOX
Slope
If You Prove... | You Can Conclude... |
Same slopes | Parallel lines |
Negative reciprocal slopes | Perpendicular lines |
Distance
If You Prove... | You Can Conclude... |
Equal side lengths | Congruent sides |
Equal diagonal lengths | Congruent diagonals |
Midpoint
If You Prove... | You Can Conclude... |
Same midpoint for both diagonals | Diagonals will bisect each other |
Diagonals bisect each other | it’s a Parallelogram |
LEVEL 7: QUADRILATERAL PROOFS
This is where midpoint, slope, distance, and diagonals all come together.
Proving a Parallelogram
Any ONE of these works and proves a parallelogram.
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Both pairs of opposite sides parallel | Parallelogram | Prove sides are parallel using Level 2: parallel lines and transversals: see if (1) corresponding angles are congruent (2) Alternate interior angles are congruent (3) Alternate exterior angles are congruent (4) Same-side interior angles are supplementary |
Both pairs opposite sides congruent | Parallelogram | check for congruent sides |
One pair opposite sides both parallel and congruent | Parallelogram | Prove sides are parallel using Level 2: parallel lines and transversals: see if (1) corresponding angles are congruent (2) Alternate interior angles are congruent (3) Alternate exterior angles are congruent (4) Same-side interior angles are supplementary |
Opposite angles congruent | Parallelogram | check for congruent angles |
Diagonals bisect each other | Parallelogram | Use midpoint formula to figure out of diagonals share the same midpoint (this is where they would intersect); if they share a midpoint (midpoint of a line is the point that splits both halves into equal pieces) and intersect at the midpoint (guaranteed if they share the same midpoint) they cut each other into equal points, thus bisecting each other. |
Coordinate Version
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Diagonals share same midpoint | Diagonals bisect each other | use midpoint formula on both diagonals to find the midpoint of both and see if they are equal/share the same midpoint |
Diagonals bisect each other | Parallelogram | (chain proof) |
Proving a Rectangle
Any ONE of these works and proves a rectangle.
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Parallelogram + one right angle | Rectangle | see Proving a Parallelogram section |
Parallelogram + congruent diagonals | Rectangle | see Proving a Parallelogram section; use distance formula to find length of diagonals if not given |
Coordinate Version
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Opposite sides parallel + adjacent sides perpendicular | Rectangle | See Level 2: parallel lines and transversals to prove if they are parallel |
Diagonals congruent and figure already a proven parallelogram | Rectangle | See Proving a Parallelogram section; use distance formula to find the length of diagonals |
Proving a Rhombus
Any ONE of these works and proves a rhombus.
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Four congruent sides | Rhombus | see if sides are congruent |
Parallelogram + perpendicular diagonals | Rhombus | see Proving a Parallelogram section; see Coordinate Geometry Toolbox: Slope to find perpendicular slope; |
Parallelogram + diagonal bisects angle | Rhombus | see Coordinate Geometry Toolbox: Midpoint |
Coordinate Version
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Diagonals perpendicular and figure is parallelogram | Rhombus | see Coordinate Geometry Toolbox: Slope and Proving Parallelogram section |
Proving a Square
Any ONE of these works and proves a parallelogram.
If You Prove... | You Can Conclude... | How to prove (which theorems or formulas to use) |
Rectangle + Rhombus | Square | see: Proving a Rectangle; Proving a Rhombus sections |
Parallelogram + congruent diagonals + perpendicular diagonals | Square | see: Proving a Parallelogram; Coordinate Geometry Toolbox: distance, slope sections |
MEDIUM-HARD PROOF QUESTIONS (1-15)
1. Proving Parallel Lines
Given:
∠1 = 4x + 10
∠2 = 6x - 20
The angles are corresponding angles.
What value of x proves the lines are parallel?
Answer
4x + 10 = 6x - 20
30 = 2x
x = 15
Proof Idea
Corresponding Angles Converse
If corresponding angles are congruent, then the lines are parallel.
2. Which Theorem?
You are given:
Which congruence theorem proves the triangles congruent?
A. SSS
B. SAS
C. ASA
D. HL
Answer
C
Proof Idea
Two angles and the included side.
ASA
3. Parallel or Perpendicular?
Line m slope = 3/4
Line n slope = -4/3
What conclusion can be made?
Answer
m ⊥ n
Proof Idea
Negative reciprocal slopes
⇒ perpendicular lines
4. Midpoint Proof
Draw on a coordinate plane and connect into its shape:
A(1,5)
C(9,1)
B(2,8)
D(8,-2)
Do diagonals AC and BD bisect each other?
Answer
Midpoint AC
((1+9)/2, (5+1)/2)
=(5,3)
Midpoint BD
((2+8)/2, (8+(-2))/2)
=(5,3)
yes they bisect each other because they share a midpoint which means if they are crossing they are guaranteed to cross at their midpoint.
Proof Idea
ABCD is a parallelogram
5. Discerning Shapes
A quadrilateral is known to be a parallelogram.
Its diagonals are congruent.
What figure must it be?
Answer
Rectangle
Proof Idea
Parallelogram + congruent diagonals
⇒ Rectangle
6. Discerning Shapes
A quadrilateral is known to be a parallelogram.
Its diagonals are perpendicular.
What figure must it be?
Answer
Rhombus
7. Similarity
DEAB=EFBC
and
∠B ≅ ∠E
Which theorem proves similarity?
Answer
SAS Similarity
8. Triangle Congruence
AB = DE
BC = EF
AC = DF
Which theorem?
Answer
SSS
9. Coordinate Parallel Proof
Plot these segments on a coordinate grid.
Segment AB:
A(1,2)
B(5,10)
Segment CD:
C(3,1)
D(7,9)
Prove AB ∥ CD
Answer
Slope AB
8/4=2
Slope CD
8/4=2
yes they are parallel because their slopes are the same
10. Coordinate Perpendicular Proof
Segment AB: A(0,0) B(2,4)
Segment CD: C(5,0) D(3,-1)
Prove segments are perpendicular.
Answer
Slope AB=2
Slope CD=1/2
Not perpendicular/cannot prove perpendicular because perpendicular lines should have negative inverse slopes, these are not negative inverse slopes.
11. Deductive Chain
Given that ABCD is a rhombus.
What must be true?
Select all.
A. Opposite sides parallel
B. Opposite angles congruent
C. Diagonals bisect each other
D. All sides congruent
Answer
All four because a Rhombus inherits every parallelogram property.
12. Discerning Shapes
A quadrilateral has:
- Congruent diagonals
- Parallel opposite sides
What figure is this?
Answer
Rectangle
13. Triangle Theorems
Two right triangles have:
Hypotenuse = 10
One leg = 6
Are these triangles congruent? How do you know?
Answer
yes, HL theorem
14. Similarity Logic
Two triangles have:
∠A ≅ ∠D
∠B ≅ ∠E
What theorem proves similarity?
Answer
AA Similarity
15. Sufficient Information
Which is sufficient to prove a quadrilateral is a parallelogram?
A. One pair opposite sides congruent
B. One pair opposite angles congruent
C. Diagonals bisect each other
D. Adjacent sides congruent
Answer
C
HARD-VERY HARD QUESTIONS (16-20)
16. Multi-Step Coordinate Proof
Given:
A(-1,0)
B(3,4)
C(7,0)
D(3,-4)
Sketch this shape on a coordinate grid and prove which type of quadrilateral this is — as specifically as you can (the most specific quadrilateral)
Answer/Explanation
Midpoint AC =(3,0)
Midpoint BD =(3,0)
:. Parallelogram
Distance AC = 8
Distance BD = 8
:. Congruent diagonals → Rectangle
Slope AC = 08 = 0
Slope BD = 0−8 = undefined
:. Perpendicular → Rhombus
:. Rectangle + Rhombus = square
Answer
Square
17. Triangle Missing Side
Answer
Scale factor
18/12=3/2
EF=12
Proof Idea
Corresponding sides proportional for similar triangles
18. Two-Theorem Transversals
A transversal intersects two lines.
Alternate interior angles measure:
5x+10 and 8x-26
Find x and state the conclusion.
Answer
5x+10=8x-26
36=3x
x=12
Angles congruent
⇒ lines parallel
19. Most Specific Quadrilateral
A quadrilateral has:
- Opposite sides parallel
- Diagonals congruent
- Diagonals perpendicular
What is the most specific classification?
A. Parallelogram
B. Rectangle
C. Rhombus
D. Square
Answer
D
Proof Chain
Parallel sides
⇒ Parallelogram
Congruent diagonals
⇒ Rectangle
Perpendicular diagonals
⇒ Rhombus
Rectangle + Rhombus
⇒ Square
20. SOL-Style Sufficient Information
A. ∠A + ∠B = ∠D + ∠E
B. ∠A ≅ ∠D
C. ∠B ≅ ∠E
D. ∠C ≅ ∠F
Answer
C
Why
AB = DE
BC = EF
Included angle B = E
⇒ SAS Congruence