The Heart of Geometry is Proofs:
Janet L Bryson
TASEL-M Orange Cluster
How to Prove:
I.
Segments Congruent
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
7. ____________________________________________________________
8. ____________________________________________________________
9. ____________________________________________________________
10. ____________________________________________________________
11. ____________________________________________________________
12. ____________________________________________________________
13. ____________________________________________________________
14. ____________________________________________________________
15. ____________________________________________________________
16. ____________________________________________________________
II.
Angles Congruent
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
7. ____________________________________________________________
8. ____________________________________________________________
9. ____________________________________________________________
10. ____________________________________________________________
11. ____________________________________________________________
12. ____________________________________________________________
13. ____________________________________________________________
Janet L Bryson
TASEL-M Orange Cluster
14. ____________________________________________________________
15. ____________________________________________________________
16. ____________________________________________________________
III.
Triangles Congruent
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
IV.
Lines Parallel
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
7. ____________________________________________________________
8. ____________________________________________________________
V.
Lines Perpendicular
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
VI.
A Quadrilateral is a Parallelogram
1. ____________________________________________________________
2. ____________________________________________________________
Janet L Bryson
TASEL-M Orange Cluster
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
VII.
Angles are Supplementary
1. ____________________________________________________________
2. ____________________________________________________________
3. ____________________________________________________________
4. ____________________________________________________________
5. ____________________________________________________________
6. ____________________________________________________________
B
F
Given : BD  FD
D is the midpoint of CE
E
D
C
BCD and FED are right angles.
Prove: BCD  FED
Statements
Reasons
1. BD  FD
1. _____________________________
2. _____________________
2. Given
3. _____________________
3. Definition of a midpoint
4.
BCD and FED are right
4. _____________________________
angles.
1. ________________ are right
5. Definition of right triangles.
triangles.
6. BCD FED
6. ________________________
Janet L Bryson
TASEL-M Orange Cluster
E
C
F
Given: AB  AE, ACB  ADE
B
A
D
Prove: B  E
Statements
Reasons
1. _____________________
1. ______________________
2. _____________________
2. Reflexive Property of
3. ABC
3. ______________________
AED
4. ___________________
4. ______________________
Prove: If a quadrilateral has one pair of sides both parallel and congruent, then the
quadrilateral is a parallelogram. (Use the definition of a parallelogram.)
B
C
Given:
Prove:
A
D
Statements
Janet L Bryson
Reasons
TASEL-M Orange Cluster
S
U
V
R
T
Use the techniques of coordinate geometry to prove that the segment joining the midpoints
of two sides of a triangle is parallel to the third side and one half its length.
2. Create coordinates for each point:
R _________ T __________
S ________
3. Find the coordinates of the midpoints U and V.
U ___________ V _________________
4. Calculate the slope of segments UV and RT.
5. Find the lengths of UV and RT.
Janet L Bryson
TASEL-M Orange Cluster
How to Prove:
I.
Segments Congruent
1. Def. of Congruent: lengths are equal
2. Def. of isosceles
3. If the 2 base angles of a triangle are congruent
4. Corresponding parts of congruent triangles
5. Opposite sides of a parallelogram
6. Radii of a circle
7. If a triangle is equiangular, it is equilateral
8. Def. of midpoint or bisect
9. Tangents to a circle from a point outside the circle
10. Chords that intercept congruent arcs of congruent or the same circle
11. Diameters perpendicular to a chord bisect the chord
12. Points on the angle bisector are equidistant from the sides of the angle
13. Diagonals of a parallelogram bisect each other
14. Def. of a square: equilateral with 4 right angles
15. Diagonals of a rectangle are congruent
16. If the base angles of a trapezoid are congruent, the trapezoid is isosceles
17. __________________________________________________________________
II.
Angles Congruent
1. If 2 lines are parallel, corresponding angles are congruent
2. If 2 lines are parallel, alternate interior angles are congruent
3. If 2 lines are parallel, alternate exterior angles are congruent
4. Vertical angles are congruent
5. 2 angles complementary to the same angle are congruent
6. 2 angles supplementary to the same angle are congruent
7. Right angles are congruent
8. If 2 sides of a triangle are congruent, the angles opposite them are congruent
9. The angles of an equilateral triangle are congruent
10. Opposite angles of a parallelogram are congruent
11. Base angles of a isosceles trapezoid are congruent
12. Corresponding parts of congruent triangles are congruent
Janet L Bryson
TASEL-M Orange Cluster
13. Def. of similar: Corresponding angles of similar triangles are congruent
14. ____________________________________________________________
III.
Triangles Congruent
1. Side-Angle-Side
2. Side-Side-Side
3. Angle-Side-Angle
4. Angle-Angle-Side
5. Hypotenuse-Leg
6. Hypotenuse-Angle
7. The diagonal of a parallelogram forms two congruent triangles.
IV.
Lines Parallel
1. Congruent corresponding angles
2. Congruent alternate interior angles
3. Congruent alternate exterior angles
4. If angles on the interior angles on the same side of the transversal are congruent
5. 2 lines parallel to the same line
6. 2 lines perpendicular to the same line
7. If a line divides 2 sides of a triangle proportionally, then it is parallel to the third side
8. Midsegment (Midline) Theorem: The segment joining the midpoints of 2 sides of a
triangle is parallel to the base and half as long
9. Def. of a parallelogram: opposite sides are parallel
V.
Lines Perpendicular
1. If they form a right angle_
2. Two angles that are congruent and adjacent
3. Converse to the Pythagorean theorem
4. A tangent is perpendicular to a radius at the point of tangency
5. A triangle inscribed in a semi-circle is a right triangle
6. Diagonals of a rhombus
7. The length of a segment is calculated as the perpendicular distance
Janet L Bryson
TASEL-M Orange Cluster
8. ____________________________________________________________
VI.
A Quadrilateral is a Parallelogram
1. Def. of a parallelogram: quadrilateral with 2 pair of opposite sides parallel.
2. Both pair of opposite sides congruent
3. Both pair of opposite angles congruent
4. 1 angle is supplementary to both consecutive angles
5. Diagonals bisect each other
6. ____________________________________________________________
VII.
Angles are Supplementary
1. If 2 angles form a linear pair
2. Given parallel lines, interior angles on the same side of the transversal
3. Given parallel lines, exterior angles on the same side of the transversal
4. Consecutive angles of a parallelogram
5. If a quadrilateral is inscribed in a circle, opposite angles are supplementary
6. Def. of supplementary: 2 angles whose measures sum to 180
7. ____________________________________________________________
B
F
Given : BD  FD
D is the midpoint of CE
E
D
C
BCD and FED are right angles.
Prove: BCD  FED
Statements
Reasons
1. BD  FD
1. Given
2. D is the midpoint of CE
2. Given
3. CD  DE
3. Definition of a midpoint
4.
4. Given
BCD and FED are right
angles.
5. BCD and FED are right
triangles.
Janet L Bryson
5. Definition of right triangles.
TASEL-M Orange Cluster
6. BCD
FED
6. HL
E
C
F
Given: AB  AE, ACB  ADE
B
A
D
Prove: B  E
Statements
Reasons
1. AB  AE, ACB  ADE
1. Given
2. A  A
2. Reflexive Property of
3. ABC
3. AAS
AED
4. B  E
4. CPCTC
Prove: If a quadrilateral has one pair of sides both parallel and congruent, then the
quadrilateral is a parallelogram. (Use the definition of a parallelogram.)
B
C
Given:
Prove:
D
A
Statements
Reasons
1. AB CD, AB  CD
1. Given
2. ABD  CDB
2.
3. BD  BD
4. ABD  CDB
5. ADB  CDB
3. Reflexive Property
4. SAS
6. AD BC
7. ABCD is a parallogram
Janet L Bryson
lines  alt. int. 's 
5. CPCTC
6. Alt. int. 's   lines
7. Def. of a parallogram: both pairs of opp sides
TASEL-M Orange Cluster
S
U
V
R
T
Use the techniques of coordinate geometry to prove that the segment joining the midpoints
of two sides of a triangle is parallel to the third side and one half its length.
6. Create coordinates for each point:
R _________ T __________
S ________
7. Find the coordinates of the midpoints U and V.
U ___________ V _________________
8. Calculate the slope of segments UV and RT.
9. Find the lengths of UV and RT.
Janet L Bryson
TASEL-M Orange Cluster
Janet L Bryson
TASEL-M Orange Cluster
How to Prove:
I.
Segments Congruent
1. Def. of Congruent: lengths are equal
2. Def. of Isosceles
3. If the 2 base angles of a triangle are congruent
4. Corresponding parts of congruent triangles
5. Opposite sides of a parallelogram
6. Radii of a circle
7. If a triangle is equiangular, it is equilateral
8. Def. of midpoint or bisect
9. Tangents to a circle from a point outside the circle
10. Chords that intercept congruent arcs of congruent or the same circle
11. Diameters perpendicular to a chord bisect the chord
12. Points on the angle bisector are equidistant from the sides of the angle
13. Diagonals of a parallelogram bisect each other
14. Def. of a square: equilateral with 4 right angles
15. Diagonals of a rectangle are congruent
16. If the base angles of a trapezoid are congruent, the trapezoid is isosceles
17. __________________________________________________________________
II.
Angles Congruent
1. If 2 lines are parallel, corresponding angles are congruent
2. If 2 lines are parallel, alternate interior angles are congruent
3. If 2 lines are parallel, alternate exterior angles are congruent
4. Vertical angles are congruent
5. 2 angles complementary to the same angle are congruent
6. 2 angles supplementary to the same angle are congruent
7. Right angles are congruent
8. If 2 sides of a triangle are congruent, the angles opposite them are congruent
9. The angles of an equilateral triangle are congruent
10. Opposite angles of a parallelogram are congruent
11. Base angles of a isosceles trapezoid are congruent
12. Corresponding parts of congruent triangles are congruent
13. Def. of similar: Corresponding angles of similar triangles are congruent
14. ____________________________________________________________
III.
Triangles Congruent
1. Side-Angle-Side
2. Side-Side-Side
3. Angle-Side-Angle
4. Angle-Angle-Side
5. Hypotenuse-Leg
6. Hypotenuse-Angle
7. The diagonal of a parallelogram forms two congruent triangles.
IV.
Lines Parallel
1. Congruent corresponding angles
2. Congruent alternate interior angles
3. Congruent alternate exterior angles
4. If angles on the interior angles on the same side of the transversal are congruent
5. 2 lines parallel to the same line
6. 2 lines perpendicular to the same line
7. If a line divides 2 sides of a triangle proportionally, then it is parallel to the third side
8. Midsegment (Midline) Theorem: The segment joining the midpoints of 2 sides of a triangle is parallel to
the base and half as long
9. Def. of a parallelogram: opposite sides are parallel
Janet L Bryson
TASEL-M Orange Cluster
V.
Lines Perpendicular
1. If they form a right angle
2. Two angles that are congruent and adjacent
3. Converse to the Pythagorean theorem
4. A tangent is perpendicular to a radius at the point of tangency
5. A triangle inscribed in a semi-circle is a right triangle
6. Diagonals of a rhombus
7. The length of a segment is calculated as the perpendicular distance
8. ____________________________________________________________
VI.
A Quadrilateral is a Parallelogram
1. Def. of a parallelogram: quadrilateral with 2 pair of opposite sides parallel.
2. Both pair of opposite sides congruent
3. Both pair of opposite angles congruent
4. 1 angle is supplementary to both consecutive angles
5. Diagonals bisect each other
6. ____________________________________________________________
VII.
Angles are Supplementary
1. If 2 angles form a linear pair
2. Given parallel lines, interior angles on the same side of the transversal
3. Given parallel lines, exterior angles on the same side of the transversal
4. Consecutive angles of a parallelogram
5. If a quadrilateral is inscribed in a circle, opposite angles are supplementary
6. Def. of supplementary: 2 angles whose measures sum to 180
7. ____________________________________________________________
Janet L Bryson
TASEL-M Orange Cluster