Geometry - 3RD Quarter - Worksheet
Geometry - 3RD Quarter - Worksheet
Geometry - 3RD Quarter - Worksheet
1. If ∆ ACT ≅ ∆ LES, complete the congruence statement or find the indicated measure.
A
a. ∠C ≅ ____ S L
b. m∠C = ____ 40
c. ____ ≅ CT
d. ____ = 9 units
11
e. ∠L ≅ ____ 5
93 C
f. m∠L = ______
g. AT ≅ _____
h. ____= 5 units 9 E
T U
2. TOM VER and mT = 83, mR=32, TM = 10, RE =14.Complete the congruence statements and
find the indicated measures
a. TO _____
b. O _____
c. mM= _____
d. OM = _____
e. VE _____
f. mE = _____
g. RV = _____
h. T _____
4. ΔQED ≅ Δ CAT , QE = 9x, ED = 4x+3, DQ = 5x+2, and AT = x+9. Find AC and CT.
5. Triangles ABC and DEF are congruent. If AB = DE, BC = EF, ∠ ABC =37 ° and
∠ EDF=39 ° , what is the measure of ∠ EFD ?
6. Triangles ABC and DEF are congruent. If AB = 19, BC = 17 and CA = 11, what is the perimeter of
triangle DEF?
7. The angles A, B and C are each 45°. The segments BS and CS extended are perpendicular to AC and
AB respectively, and ÁS bisects ∠ BAC. If AS = 13 cm, what is the distance B to C?
B
A C
Activity 3: Proving Congruent Triangles
B E
C D
Prove: MN JN
L K
J
M
3. Prove the Isosceles Triangle Theorem and its converse.
J K
5. Given: TM TA ,M SM RA
L
Prove: ∠S∠R
N
T
S
M A R
6. Given ∆ ABC , AB= AC .F is on side AC and CF=BD. E is the intersection of FD and BC. Prove that
FE=ED
A
B
C
E
D
II. How many pairs of congruent triangles are there? Identify the congruent triangles and support each
conclusion with the corresponding postulates.
O E
D
B
´ ∥ CD,
2. As shown in the figure, AB ´ AC
´ ∥ BD.
´ Two diagonals AD and BC meet at O. AE ⊥ BC at E and
DF ⊥ BC at F.
B
A F
O
E D
C
3. As shown in the figure, in AB = BC = CA and AD = BE = CF. None of D, E, F is the midpoint of any side.
AE, BF and CD intersect at M, N and P, respectively.
D
P
F
M N
B C
E
4. G is the centroid of isosceles Δ ABC where AB = AC.
A
D E
G
B C
F
1. Each side of the square ABCD has length 1 and m∠ PAQ=45° . What is the perimeter of Δ PQC
A
? D
B Q C
A B
3. ∆ ABC is equilateral with perimeter 3 units. ∆ BDC is isosceles with DB = DC and m∠ BDC=120 ° . If
points E and F are on AB and AC respectively and m∠ EDF =60 °, what is the perimeter of ∆ AEF ?
A
B C
D
Activity 4 Right Triangle Congruence
A C
E B D
C
3. Given: BC AD, A D
D ABC DBC
Prove:
C B
D E
B
E
C D
A
6. Given: AC ⊥ DB at O, DB and AC bisect each other.
D
Prove: ∆ AOD ≅ ∆COB
O
B
7. In triangle ABC, BD = CE, DM = ME. Prove that triangle ABC is an isosceles triangle.
A
B C
M
8. Given isosceles right triangle ABC with AB as the hypotenuse. D is the midpoint of BC and CE is
perpendicular to AD. Prove that ∠ CDF ≅ ∠BDE.
E
F
B
C D
III. Solve the following problems.
1. Solve the following. For numbers 1 to 10 refer to the given figure below.
CD AB, BE AC A
CD BE, BD = 5x –7
CE = 2x + 14, DF = 2x + 5, E
D
EF = 3x – 2.
1. Find x. F
2. Find BD
3. BDC CEB by _____ theorem. B C
4. Find DF
5. DFB EFC by _____ theorem.
CD AB, BE AC
AC AB, AD = 4x –5
AE = 2x + 7, mA = 3x +8.
6. Find x.
7. Find AD
8. Find mA
9. Find mC
10. AEB ADC by _____ theorem.
2. Let ABCD be a square. P is on side AB with AP = 2BP. Point Q is on side BC with BQ = 2CQ. What is
the sum of the measures of the angles ∠QAB , ∠ PDQ and ∠ PCB .
D C
A P B
E
F
B C
D
4. In Δ ABC , m∠ ACB=60 ° , m∠ BAC=75 ° , AD ⊥ BC at D, BE ⊥ AC at E and AD and
BE meet at H. Find m∠CHD .
A
H E
B D C
B C
D
6. ∆ ABC is equilateral. Points D and E are on AC and AB respectively. BD and CE intersect at F. If the
area of quadrilateral ADFE is equal to the area of ∆ BFC. Find the measure of ∠ BFE .
D
F
B C
Activity 5 Triangle Inequalities Contest Round
a. 3, 8, 12
b. 4, 7, 9
c. 3, 5, 1
d. 6, 15, 17
4. If A, B and D are three points such that G – K – H, which of the following is always true?
a. GK = KH
b. GK < KH
c. GK > GH
d. KH < GH
a. 11 b. 15 c. 21 d. 27
8. Given the parcel of land illustrated by the figure below, give the inequality that describes the possible
values of x.
4x – 4
28
115°
140°
9. Given the piece of farming lot illustrated by the figure below, give the inequality that describes the
possible values of x.
(3x – 9)°
36°
11
11
10. From the park, Jam rides his bike due north for 3 kilometers, then turns N 120° W for 1.25 kilometers.
Jepson leaves the park and rides his bike due south for 3 kilometers then turns due east for 1.25
kilometers. Which biker is now farther from the park?
60°
P 46°
90° 25°
M K
12. In the figure below which angle is the largest acute angle?
K
15 13
12
G 9 M 5H
13. DJ has two bamboo sticks with measures 32cm and 19cm, if he will make a triangular picture frame,
how many possible picture frames can he make, if the third side has integral length?
14. In the figure, what is the range of possible values of AC?
A 4 B
6 8.5
D C
11
15. A 144 inch piece of wire is bent to form a triangle with integral lengths. How many isosceles triangles
can be formed?
Activity 1: Axiomatic Structure of Geometry
Define the following terms inside the box and come up with a concept map to show the interrelationship
between the different terms.
3. AD≃CD , m∠ BDC=110°
Prove: CB > AB
A C
D
4. In ∆ ABC, AB= AC; D is any point of BC ´ not on BC´ . Prove that DA > AC.
5. Prove that the sum of the distances from a point in the interior of a triangle to the ends of one side is
less than the sum of the lengths of the other two sides of the triangle.