Unit 1 Main Points
General Properties:
Reflexive Property
Substitution Postulate
Addition Postulate
Subtraction Postulate
Angle Theorems:
Right Angles
Linear Pair
Vertical Angles
Triangle Sum
Base Angle Theorem
(Isosceles Triangle)
Base Angle Converse
(Isosceles Triangle)
Triangles:
Triangle Congruence
Triangle Similarity
CPCTC
A quantity is congruent (equal) to itself. Usually used with a shared side.
A quantity may be substituted for its equal in any expression.
You can add the same thing to both sides of an equation.
You can subtract the same thing from both sides of an equation.
All right angles are congruent.
If two angles form a linear pair, they are supplementary.
Vertical angles are congruent.
The angles in a triangle add to 180°.
If two sides of a triangle are congruent, the angles opposite these sides are
congruent.
If two angles of a triangle are congruent, the sides opposite these angles are
congruent.
SSS, SAS, ASA, AAS, HL
AA
SSS, SAS
The triangles are the same shape and size, so their corresponding parts are
congruent.
Parallel Lines:
Corresponding angles are congruent.
Alternate interior angles are congruent.
Converse: If these angles are congruent, the lines
are parallel.
Alternate exterior angles are congruent.
Same side interior (consecutive interior) angles are
supplementary.
1-21 Triangle Proportionality Theorem: If a line parallel to one side of a triangle
intersects the other two sides, then it divides those sides proportionally.
𝐷𝐺
𝐺𝐸
𝐺𝐸
𝐷𝐸
𝐷𝐺
𝐷𝐸
𝐺𝐸
𝐻𝐹
=
=
=
=
𝐷𝐻
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝐻𝐹
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝐻𝐹
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝐷𝐹
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
𝐷𝐻
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝐷𝐹
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
𝐷𝐺
𝐷𝐻
=
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
=
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
=
=
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
𝐷𝐸
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝐷𝐹
𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
=
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
𝑙𝑎𝑟𝑔𝑒𝑟 𝑝𝑖𝑒𝑐𝑒
=
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
𝑤ℎ𝑜𝑙𝑒 𝑠𝑖𝑑𝑒
1-23 Triangle Mid-Segment Theorem
The Triangle Mid-Segment Theorem is a special case of the
Triangle Proportionality Theorem.
Triangle Mid-Segment Theorem: If a segment joins the
midpoints of two sides of a triangle, then the segment is
parallel to the third side and half its length.
1-22 Triangle Angle-Bisector Theorem: If a segment bisects an angle of a triangle, then it divides the
opposite side into segments proportional to the other two sides.
Exterior angle theorem:
Unit 1 EOCT Practice Questions
1. In the diagram of ΔABC below, AB = 10, BC = 14, and
AC = 16. Find the perimeter of the triangle formed by
connecting the midpoints of the sides of ΔABC.
2. Trapezoid TRAP is shown below.
3. Which illustration shows the correct construction of an
angl bisector?
4. Which diagram shows the construction of an
equilateral triangle?
5. Line segment AB is shown in the diagram below.
6. Which geometric prniciple is used in the
construction shown below?
7. Given: Quadrilateral ABCD,
, and
8. The statements for a proof are shown.
Reason
Given: Parallelogram ABCD
∠1≅∠2
1.
1. Given
Prove: 𝐵𝑋 ≅ 𝐷𝑌
2.
2. Given
3.
3. Reflexive Property
Statement
4.
4. SSS Postulate
5.
5. CPCTC
6. ?
6.
7.
Statements
1. Parallelogram ABCD
∠1≅∠2
2. ∠ B ≅ ∠ D
3. 𝐴𝐵 ≅ 𝐷𝐶
4. Δ ABX ≅ Δ CDY
5. 𝐵𝑋 ≅ 𝐷𝑌
9. ABCD is a parallelogram
8. ?
9. Definition
In the proof shown above, which postulate or
theorem justifies steps 6 and 8?
What is the reason that the statement in Step 5 is true?
A. angle-side-angle
B. side-angle-side
C. Opposite sides of a parallelogram are congruent.
D. Corresponding parts of congruent triangles are
congruent.
A. CPCTC
B. Corresponding angles theorem
C. Alternate interior angles theorem
D. Definition of parallel lines
9. Complete the problems - solve for x.
d)
2.
3.
4.
5.
7. CPCTC
8.
a)
Reasons
1. Given
b)
c)
10. Given: PQRS is a parallelogram; S, P, and T are collinear.
Prove: 1 4.
Statements
1. PQRS is a
parallelogram; S, P, and
T are collinear
Reasons
1. Given
2.
2. Definition of a
parallelogram
3. Parallel lines cut by a
transversal form congruent
corresponding angles
4. Definition of a
parallelogram
5. ?
6. Transitive property of
congruence
3.
1
3
4
1
3
4
4.
5.
6.
11. Manuel is trying to prove the following theorem. If
two sides of a triangle are congruent, then the angles
opposite these sides are congruent. First Manuel draws
isosceles
and then he adds an auxiliary line that
bisects
An incomplete version of Manuel’s proof is shown below.
Which reason could be used to justify Statement 5?
A. Angles that are supplements of the same angle are congruent
to each other.
B. Parallel lines cut by a transversal form congruent alternate
interior angles.
C. Parallel lines cut by a transversal form congruent
corresponding angles.
D. Intersecting lines form congruent vertical angles.
What should be the statement for Step 4 of Manuel’s
proof?
A.
is a right angle
B.
C.
D.
12.
Statements
GIVEN: Isosceles trapezoid ABCD with bases
and
Reasons
1. Isosceles trapezoid ABCD
1. Given
2.
2. Definition of isosceles trapezoid
3.
3. Reflexive property
4.
4. ?
5.
5. SAS postulate
6.
6. Corresponding parts of
congruent triangles are congruent
PROVE:
Which is the reason for Step 4?
A. The diagonals of an isosceles trapezoid are congruent.
B. Both pairs of base angles of an isosceles trapezoid are congruent.
C. If two parallel lines are cut by a transversal, corresponding angles
are congruent.
D. If two parallel lines are cut by a transversal, alternate interior
angles are congruent.
13. GIVEN: Parallelogram WXYZ
PROVE:
bisects
,
bisects
Which could be the statement for Step 7?
A.
,
B.
C.
,
D.
,