SMG3013

GEOMETRY

SEMESTER 2
SESSION 2019/2020

ASSIGNMENT 1

PREPARED BY:

NO. NAME MATRIC ID

1. KOH KAI LING @ NORHALINDA A/P KOH TIM WEON D20162076379
2. NURUL SYAZWANI BINTI MOHD ZID D20162076380
3. NURSHAFAZLEEN BINTI AK DAMIT D20162076387
4. MOHAMMAD FARID ASYRAAF BIN MOHD YAINI D20182086632

GROUP: A
LECTURER’S NAME: DR. NOR SURIYA BINTI ABD KARIM
SUBMISSION DATE: 28 JUNE 2020
Part A (Question 1)

NO. OBJECT CHARACTERISTIC

1) Right angle

2) Obtuse angle

3) Isosceles sides

4) Equilateral sides

5) Equiangular angle
Part A (Question 2)

Usage of the Party flag are use in birthday

Object party or event

Size

65o 65o measurement

- Isosceles Sides

Angles
o
50
measurement

- Acute Angle

Type of triangle Isosceles triangle

Part B (Question 1a)

Two triangles are said to be similar if their corresponding angles are congruent and the

corresponding sides are in proportion. In other words, similar triangles are the same shape

but not necessarily the same size. The triangles are congruent, if, in addition to this, their

corresponding sides are of equal length. Referring Theorem 5.1.1, the lengths of the

corresponding altitudes of similar triangles have the same ratio as the lengths of any pair

of corresponding sides.

Think of it as ‘zooming in’ or out making the triangle bigger or smaller, but keeping

its basic shape. Two triangles can be similar, even if they have share some elements. In the

figure below, the larger triangle PQR is similar to the smaller one STR. S and T are the

midpoints of PR and QR respectively. They share the vertex R and part of the sides PR and

QR. They are similar on the basis of AAA, since the corresponding angles in each triangle

are the same.

Part B (Question 1b)

Similar triangles are very useful for indirectly determining the sizes of items which are

difficult to measure by hand. Similar triangles may show up everywhere in real life even if

we are unable to notice them at first. Typical examples are including building heights, tree
heights or tower heights. Similar triangles can also be used to measure how wide a river or

a lake is.

The use of similar triangles is of outmost importance where it is beyond our reach

to physically measure the distances and heights with simple measuring instruments. The

uses of similar triangles thus can influence the varied number of fields. This numerous

applications are majorly in the fields of engineering, architecture and construction.

i) To analyze the stability of bridges. Stability is an important factor when

something is build.

ii) In real life projects, similar triangles are used to hold the ground when an

earthquake arises.

iii) In aerial photography, similar triangles are to determine the distances from

sky to a particular point on ground.

Part B (Question 2a)

Methods for proving SIMILAR triangles:

1. AAA

2. AA

3. CSSTP

4. CASTC

5. SAS~

6. SSS~

Method Statement

If in two triangles, the corresponding angles are equal,

AAA
i.e., if the two triangles are equiangular, then the triangles are similar.

If two angles of one triangle are respectively equal to two angles of

AA
another triangle, then the two triangles are similar.

CSSTP Corresponding Sides of Similar Triangles Are Proportional

CASTC Corresponding Angles of Similar Triangles Are Congruent

If in two triangles, one pair of corresponding sides are proportional and

SAS~
the included angles are equal then the two triangles are similar.

If the corresponding sides of two triangles are proportional, then the

SSS~
two triangles are similar.
Part B (Question 2b)

Method Example of Proof

Triangles ABC and DEF such that ∠𝐴 = ∠𝐷; ∠𝐵 = ∠𝐸; ∠𝐶 = ∠𝐹.

Prove that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹.

Construction : We mark point P on the line DE and Q on the line DF

such that AB=DP and AC=DQ, we join PQ

Case (i) : AB = DE, thus P coincides with E

Statement Reason
1) AB = DE According to 1st case
AAA
2) ∠𝐴 = ∠𝐷 Given

3) ∠𝐵 = ∠𝐸 Given

4) ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 By ASA postulate

5) AB = DE; BC = EF; Substitution

CA = FA

6) ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 Proven

D is a point on the side of BC of ∆𝐴𝐵𝐶 such that ∠𝐴𝐷𝐶 = ∠𝐵𝐴𝐶. Prove

AA that CA2 = BA × CD

Statements Reason
 1) ∠𝐴𝐷𝐶 = ∠𝐵𝐴𝐶 Given

2) ∠𝐶 = ∠𝐶 Reflexive

3) ∆𝐴𝐵𝐶~∆𝐷𝐴𝐶 AA criteria

𝐴𝐵 𝐶𝐵 𝐶𝐴 If two triangles are similar then their

4) 𝐷𝐴 = 𝐶𝐴 = 𝐶𝐷
sides are in proportion

𝐶𝐵 𝐶𝐴
Last two ratios
5) 𝐶𝐴 = 𝐶𝐷

6) CA2 = BA × CD
Cross Multiplication (proven)

̅̅̅̅ . Prove JL∘ 𝑁𝑃 = 𝑄𝑁 ∘ 𝐿𝐾

∆𝐿𝑀𝑁 is isosceles with base 𝐿𝑁

Statements Reason
1) ∆𝐿𝑀𝑁 is isosceles with △ Given

̅̅̅̅ . ∠1 ≅ ∠8
with base 𝐿𝑁

̅̅̅̅ ≅ 𝑀𝑁
2) 𝑀𝐿 ̅̅̅̅̅ Definition of isosceles triangle

3) ∠4 ≅ ∠5 If sides, then angles

4) ∠3 ≅ ∠4 Vertical angles are congruent

CSSTP
∠5 ≅ ∠6

5) ∠3 ≅ ∠6 Transitive Property for four angles

(If two angles are congruent to two

other congruent angles, then they’re

congruent) (Given)

6) ∠1 ≅ ∠8 Supplements of congruent angles are

congruent
 7) ∠2 ≅ ∠7 AA

8) ∆𝐽𝐾𝐿~∆𝑄𝑃𝑁 CSSTP

𝐽𝐿 𝐿𝐾 Cross-multiplication
9) 𝑄𝑁 = 𝑁𝑃
Proven
10) JL∘ 𝑁𝑃 = 𝑄𝑁 ∘ 𝐿𝐾

̅̅̅̅. Prove
Given ∆𝑅𝑆𝑇 is a right triangle with right angle ∠𝑆, and 𝑇𝑅

∆𝑅𝑆𝑇~∆𝑅𝑉𝑆, ∆𝑅𝑉𝑆~∆𝑆𝑉𝑇, and ∆𝑆𝑉𝑇~∆𝑅𝑆𝑇.

Statement Reason
̅̅̅̅ is the altitude to 𝑇𝑅
1. 𝑆𝑉 ̅̅̅̅ Given

̅̅̅̅ ⊥ 𝑇𝑅
2. 𝑆𝑉 ̅̅̅̅ An altitude is a segment from a vertex

and perpendicular the opposite side

3. ∠𝑅𝑉𝑆 and ∠𝑆𝑉𝑇 are right Angles formed by perpendicular lines

CASTC
angles are perpendicular

4. ∠𝑆 is a right angle Given

5. ∠𝑅𝑉𝑆 ≅ ∠𝑆𝑉𝑇 ≅ ∠𝑆 Right angles are congruent

6. ∠𝑉 ≅ ∠𝑉 Identity

7) ∆𝑅𝑆𝑇~∆𝑅𝑉𝑆 If two angles of one triangle are

congruent to two angles of another

triangle, then the triangles are similar

(AA)

8) ∠𝑇 ≅ ∠𝑇 Identity
 9) ∠𝑆𝑅𝑉 ≅ ∠𝑇 Corresponding angles of similar

triangles are congruent (CASRC)

10) ∆𝑅𝑉𝑆~∆𝑆𝑉𝑇 If two angles of one triangle are

congruent to two angles of another

triangle, then the triangles are similar

(AA)

11) ∠𝑆𝑇𝑉 ≅ ∠𝑅 Corresponding angles of similar

triangles are congruent (CASRC)

12) ∠𝑅 ≅ ∠𝑅 Identity

13) ) ∆𝑆𝑉𝑇~∆𝑅𝑆𝑇 If two angles of one triangle are

congruent to two angles of another

triangle, then the triangles are similar

(AA)

Given two triangles ABC and DEF such that ∠𝐴 = ∠𝐷, AB = DE; AC

= DF. Prove that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹. (Construction: Let P and Q be two

points on DE and DF respectively such that DP = AB and DQ = AC.

Join PQ)

Statements Reasons
SAS~
1) AB = DE; AC = DF Given

2) DP = DE; DQ = DF By substitution

3) AB = DP; ∠𝐴 = ∠𝐷 and Given and by construction

AC = DQ

4) ∆𝐴𝐵𝐶 ≅ ∆𝐷𝑃𝑄 By SAS postulate

 5) PQ || EF By converse of basic proportionality

theorem

6) ∠𝐷𝑃𝑄 = ∠𝐸 and ∠𝐷𝑄𝑃 = Corresponding angles

∠𝐹

7) ∆𝐷𝑃𝑄 ≅ ∆𝐷𝐸𝐹 By AAA similarity

8) ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 Proven

Given two triangles ABC and DEF such that AB=DE; BC=EF; CA=FD.

Prove that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹. (Let P and Q be two points on DE and DF

respectively such that DP = AB and DQ = AC. Join PQ)

Statements Reasons
1) AB = DE; Given

AC = DF

2) DP = DE; As AB = DP and AC = DQ

DQ = DF (substitution)

3) PQ || EF By converse of
SSS~
basic proportionality theorem

4) ∠𝐷𝑃𝑄 = ∠𝐸 and ∠𝐷𝑄𝑃 = Corresponding angles

∠𝐹

5) ∆𝐷𝑃𝑄~∆𝐷𝐸𝐹 By AA similarity

6) DP = DE; PQ = EF By definition of similar triangles

7) AB = DE; PQ = EF As DP = AB (substitution)

8) PQ = EF; BC = EF From (1)(6) and (7)

9) PQ = BC From (8)
 10) ∆𝐴𝐵𝐶 ≅ ∆𝐷𝑃𝑄 By SSS postulate

11) ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 Proven

Part B (Question 2c)

METHODS IN PROVING PARTS

Angle – Angle (AA) – Two angles of Corresponding Sides

one triangle are congruent to two angles of Similar Triangles

of another triangle. are proportional

Angle – Angle – Angle (AAA) – Three (CSSTP)

triangles of one triangle are congruent to

SIMILAR three angles of another triangle.

TRIANGLES Side – Angle – Side (SAS~) – Two

corresponding sides of two triangles are

proportional and the included angles are Corresponding Angles

congruent. of Similar Triangle are

Side – Side – Side (SSS~) - The three congruent (CASTC)

sides of two triangles are proportional.

 References

AA Similarity. (n.d.) Retrieved June 23, 2020, from

https://www.ask-math.com/AAsimilarity.html

AAA Similarity. (n.d.) Retrieved June 23, 2020, from

https://www.ask-math.com/AAAsimilarity.html

Geralyn M Koeberlein, Daniel C Alexander.(2008). Elementary Geometry for College

Students. Retrived from https://www.chegg.com/homework-help/elementary

geometry-for-college-students-5th-edition-chapter-5.4-solutions-9781439047903

Mark Ryan. (n.d.). How to Solve a CSSTP Proof. Retrieved June 23, 2020, from

https://www.dummies.com/education/math/geometry/solve-csstp-proof/

Pierce, Rod. (18 Jan 2020). "Triangles - Equilateral, Isosceles and Scalene". Math Is Fun.

Retrieved 22 Jun 2020 from http://www.mathsisfun.com/triangle.html

Pierce, Rod. (24 Feb 2017). "Similar Triangles". Math Is Fun. Retrieved 22 Jun 2020 from

http://www.mathsisfun.com/geometry/triangles-similar.html

SAS Similarity. (n.d.) Retrieved June 23, 2020, from

https://www.ask-math.com/SASsimilarity.html

SSS Similarity. (n.d.) Retrieved June 23, 2020, from

https://www.ask-math.com/SSSsimilarity.html

Wang, S. (2015). Identifying Similar Polygons: Comparing Prospective Teachers’

Routines with a Mathematician’s.

Weisstein, Eric W. "Equilateral Triangle." From MathWorld--A Wolfram Web

Resource. https://mathworld.wolfram.com/EquilateralTriangle.html