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Transcript
Year 7 Angles – Exercise 1
1. Find the angles marked with letters.
2.
[JMC 2000 Q3] What is the value of
π‘₯?
7.
[IMC 2011 Q9] In the diagram, π‘‹π‘Œ is a
straight line. What is the value of π‘₯?
3.
[JMC 2015 Q6] What is the value of π‘₯
in this triangle?
8.
[JMC 2010 Q8] In a triangle with
angles π‘₯°, 𝑦°, 𝑧° the mean of 𝑦 and 𝑧 is
π‘₯. What is the value of π‘₯?
4.
[JMC 2011 Q6] What is the sum of
the marked angles in the diagram?
9.
[JMC 1999 Q13] In the diagram,
βˆ π‘…π‘ƒπ‘€ = 20° and βˆ π‘„π‘€π‘ƒ = 70°.
What is βˆ π‘ƒπ‘…π‘†?
5.
6.
[JMC 1997 Q17] How big is angle π‘₯?
[IMC 2014 Q3] An equilateral triangle
is placed inside a larger equilateral
triangle so that the diagram has three
lines of symmetry. What is the value
of π‘₯?
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10. [JMC 2001 Q18] Triangle 𝑃𝑄𝑅 is
equilateral. Angle 𝑆𝑃𝑅 = 40°, angle
𝑇𝑄𝑅 = 35°. What is the size of the
marked angle 𝑆𝑋𝑇?
11. [JMC 2007 Q16] What is the sum of
the six marked angles?
12. [JMC 2015 Q16] The diagram shows a
square inside an equilateral triangle.
What is the value of π‘₯ + 𝑦?
13. [JMO 2001 A3] What is the value of π‘₯
in the diagram alongside?
14. [Kangaroo Pink 2010 Q9] In the
diagram, angle 𝑃𝑄𝑅 is 20°, and the
reflex angle at 𝑃 is 330°.The line
segments 𝑄𝑇 and π‘†π‘ˆ are
perpendicular. What is the size of
angle 𝑅𝑆𝑃?
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15. [Kangaroo Pink 2006 Q8] The circle
shown in the diagram is divided into
four arcs of length 2, 5, 6 and π‘₯ units.
The sector with arc length 2 has an
angle of 30° at the centre. Determine
the value of π‘₯.
(Hint: An arc is a part of the line that makes up
the circle, i.e. part of the circumference. The
arc length grows in proportion to the angle at
the centre)
16. [SMC 2010 Q3] The diagram shows
an equilateral triangle touching two
straight lines. What is the sum of the
four marked angles?
Exercise 2 – Z/F/C angles and quadrilaterals
1. For each diagram (i) Find the missing angle(s) and (ii) List reasons for each answer.
a)
b)
2. Find the indicated unknown angles.
3. For rhombuses are evenly spaced around a point. Given the angle shown, find π‘₯.
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4. Find the angles indicated.
10. [IMC 2005 Q7] In the diagram, what
is the sum of the marked angles?
5. Find the angle 𝛼.
11. [JMO 2003 A9] Find the value of π‘Ž +
𝑏 + 𝑐.
6. The line 𝑃𝐡 bisects the angle 𝐴𝐡𝐢
(this means it cuts it exactly in half).
Determine the angle 𝐡𝑃𝐢.
7.
[JMC 2006 Q7] What is the value of
π‘₯?
8.
[JMC 2007 Q9] In the diagram on the
right, 𝑆𝑇 is parallel to π‘ˆπ‘‰. What is the
value of π‘₯?
9.
[JMC 2009 Q19] The diagram on the
right shows a rhombus 𝐹𝐺𝐻𝐼 and an
isosceles triangle 𝐹𝐺𝐽 in which 𝐺𝐹 =
𝐺𝐽. Angle 𝐹𝐽𝐼 = 111°. What is the
size of angle 𝐽𝐹𝐼?
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12. [SMC 2006 Q3] The diagram shows
overlapping squares. What is the
value of π‘₯ + 𝑦?
13. [Cayley 2012 Q2] In the diagram, 𝑃𝑄
and 𝑇𝑆 are parallel.
Prove that π‘Ž + 𝑏 + 𝑐 = 360.
14. [Kangaroo Grey 2005 Q20] Five
straight lines intersect at a common
point and five triangles are
constructed as shown. What is the
total of the 10 angles marked in the
diagram?
A 300°
B 450°
C
360°
D 600°
E
720°
Exercise 3 – Isosceles Triangles
1. Find the value of each variable.
2. [JMC 2014 Q10] An equilateral triangle is surrounded by three squares, as shown.
What is the value of π‘₯?
3. [JMC 2005 Q13] The diagram shows two equal squares. What is the
value of π‘₯?
4. [IMC 2010 Q6] In triangle 𝑃𝑄𝑅, 𝑆 is a point on 𝑄𝑅 such that
𝑄𝑆 = 𝑆𝑃 = 𝑃𝑅 and βˆ π‘„π‘ƒπ‘† = 20°. What is the size of βˆ π‘ƒπ‘…π‘†?
5. [IMC 1999 Q8] In the diagram 𝑃𝑄 = 𝑃𝑅 = 𝑅𝑆.
What is the size of angle π‘₯?
6. [JMC 2008 Q19] In the diagram on the right, 𝑃𝑇 = 𝑄𝑇 = 𝑇𝑆, 𝑄𝑆 = 𝑆𝑅,
βˆ π‘ƒπ‘„π‘‡ = 20°. What is the value of π‘₯?
7. [Kangaroo Grey 2010 Q9] The diagram shows a quadrilateral 𝐴𝐡𝐢𝐷, in
which 𝐴𝐷 = 𝐡𝐢, ∠𝐢𝐴𝐷 = 50°, ∠𝐴𝐢𝐷 = 65° and ∠𝐴𝐢𝐡 = 70°. What is
the size of ∠𝐴𝐡𝐢?
8. [Cayley 2004 Q1] The β€œstar” octagon shown in the diagram is beautifully
symmetrical and the centre of the star is at the centre of the circle. If angle
𝑁𝐴𝐸 = 110°, how big is the angle 𝐷𝑁𝐴?
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9. [IMC 2003 Q8] Lines 𝐴𝐡 and 𝐢𝐷 are parallel and 𝐡𝐢 = 𝐡𝐷. Given that π‘₯ is
an acute angle not equal to 60°, how many other angles in this diagram are
equal to π‘₯?
10. [IMC 1997 Q11] In the quadrilateral 𝐴𝐡𝐢𝐷, ∠𝐴𝐡𝐢 = 90°, ∠𝐡𝐴𝐷 = 70°, and
𝐴𝐡 = 𝐡𝐷 = 𝐡𝐢. What is the size of ∠𝐡𝐷𝐢?
11. [Kangaroo Pink 2013 Q9] The diagram shows an equilateral triangle 𝑅𝑆𝑇
and also the triangle π‘‡π‘ˆπ‘‰ obtained by rotating triangle 𝑅𝑆𝑇 about the
point 𝑇. Angle 𝑅𝑇𝑉 = 70°. What is angle 𝑅𝑆𝑉?
12. [IMC 2000 Q8] In the triangle 𝐴𝐡𝐢, 𝐴𝐷 = 𝐡𝐷 = 𝐢𝐷. What is the size of angle
𝐡𝐴𝐢?
13. In the diagram 𝐴𝐡 is parallel to 𝐢𝐷 and 𝐴𝐸 = 𝐴𝐢.
Determine ∠𝐢𝐴𝐸.
14. [TMC Regional 2008 Q9] If 𝐴𝐡 = 𝐡𝐢 = 𝐢𝐷 =
𝐷𝐸 = 𝐸𝐹 and angle 𝐴𝐸𝐹 = 75°, what is the size of
angle 𝐸𝐴𝐹?
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Exercise 4 – Algebraic Angles
1. Given the value of ∠𝐷𝐡𝐴 indicated, work out the value of ∠𝐷𝐡𝐢.
e.g. ∠𝐷𝐡𝐴 = 𝑦
οƒ  ∠𝐷𝐡𝐢 = 180 βˆ’ 𝑦
a. ∠𝐷𝐡𝐴 = 𝑀
b. ∠𝐷𝐡𝐴 = π‘₯ + 10
c. ∠𝐷𝐡𝐴 = 2π‘₯ βˆ’ 30
d. ∠𝐷𝐡𝐴 = 90 + 2π‘₯
e. ∠𝐷𝐡𝐴 = 130 + π‘₯ βˆ’ 2𝑦
2. Given the values of ∠𝐴𝐡𝐢 and ∠𝐡𝐴𝐢 indicated, work out the value of ∠𝐴𝐢𝐡.
e.g. ∠𝐴𝐡𝐢 = 30°, ∠𝐡𝐴𝐢 = 40° οƒ  ∠𝐴𝐢𝐡 = 110°
a. ∠𝐴𝐡𝐢 = π‘₯, ∠𝐡𝐴𝐢 = π‘₯
b. ∠𝐴𝐡𝐢 = π‘₯, ∠𝐡𝐴𝐢 = 10°
c. ∠𝐴𝐡𝐢 = π‘₯ + 10°, ∠𝐡𝐴𝐢 = 150°
d. ∠𝐴𝐡𝐢 = 2π‘₯ + 30°, ∠𝐡𝐴𝐢 = 20° βˆ’ 5π‘₯
e. ∠𝐴𝐡𝐢 = 4π‘₯ βˆ’ 20°, ∠𝐡𝐴𝐢 = 7π‘₯ βˆ’ 30°
3. Determine all the angles in each diagram in terms of π‘₯ and/or 𝑦.
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4. Write suitable expressions in terms of π‘₯ for each missing angle. Hence determine the
smallest angle in the diagram in each case.
a. Angle 𝐴𝐡𝐷 is 3 times bigger than angle 𝐷𝐡𝐢.
.
b. Angle 𝐡𝐢𝐴 is twice as large as angle 𝐡𝐴𝐢.
c. The angles 𝐴𝐢𝐡, 𝐴𝐡𝐢 and 𝐡𝐴𝐢 are in the ratio 3: 4: 5.
5. [Based on JMO 2005 B4] In this figure 𝐴𝐷𝐢 is a straight line and 𝐴𝐡 = 𝐡𝐢 = 𝐢𝐷. Also, 𝐷𝐴 =
𝐷𝐡. We wish to find the size of ∠𝐡𝐴𝐢. Let ∠𝐷𝐴𝐡 = π‘₯.
a) By using the information provided, determine all the remaining angles in the diagram in
terms of π‘₯.
b) By considering the three angles in Δ𝐷𝐢𝐡, hence determine ∠𝐡𝐴𝐢.
6. [Based on JMC 2011 Q23] The points 𝑆, 𝑇, π‘ˆ lie on the sides of the triangle
𝑃𝑄𝑅, as shown, so that 𝑄𝑆 = π‘„π‘ˆ and 𝑅𝑆 = 𝑅𝑇. βˆ π‘‡π‘†π‘ˆ = 40°.
a) Let βˆ π‘‡π‘†π‘… = π‘₯° and βˆ π‘ˆπ‘†π‘„ = 𝑦°. What is the value of π‘₯ + 𝑦?
b) Using the information provided, find expressions for βˆ π‘‡π‘…π‘† and βˆ π‘ˆπ‘„π‘†.
c) Hence find an expression for βˆ π‘…π‘ƒπ‘„.
d) Using your answer to part (a), hence find the value of βˆ π‘…π‘ƒπ‘„.
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7. [Based on TMC Final 2014 Q4] The triangle 𝐴𝐡𝐢 is isosceles with 𝐴𝐢 = 𝐡𝐢 as shown.
The point 𝐷 lies on the line 𝐡𝐢 such that the triangle 𝐴𝐡𝐷 is isosceles with 𝐴𝐡 = 𝐡𝐷
as shown. ∠𝐡𝐴𝐢 = 2 × βˆ π΅πΆπ΄ and we wish to work out the value of ∠𝐴𝐷𝐢.
Suppose we let ∠𝐡𝐢𝐴 = π‘₯. Hence determine:
a) ∠𝐡𝐴𝐢
b) ∠𝐢𝐡𝐴
c) Hence by considering the triangle 𝐢𝐴𝐷, determine π‘₯.
d) Use the remaining information to determine ∠𝐴𝐷𝐢.
8. [JMO 2004 B1] In the rectangle ABCD, M and N are the midpoints of AB and CD respectively;
AB has length 2 and AD has length 1.
Given that ∠𝐴𝐡𝐷 = π‘₯°, calculate βˆ π·π‘π‘ in terms of π‘₯.
9. [Based on JMO 2006 B3] In this diagram, Y lies on the line AC, triangles
ABC and AXY are right angled and in triangle ABX, AX = BX. The line
segment AX bisects angle BAC and angle AXY is seven times the size of
angle XBC.
Let βˆ π‘‹π΅πΆ = π‘₯. In terms of π‘₯, find expressions for:
a) βˆ π΄π‘‹π‘Œ
b) βˆ π΅π‘‹π‘Œ (hint: the lines π‘‹π‘Œ and 𝐡𝐢 are parallel)
c) βˆ π‘‹π΄π‘Œ
d) βˆ π΅π΄π‘‹
e) βˆ π΅π‘‹π΄ (using the fact that Δ𝐡𝑋𝐴 is isosceles)
f) Optional: By considering angles around a suitable point, hence find ∠𝐴𝐡𝐢.
10. [JMC 2003 Q23] In the diagram alongside, 𝐴𝐡 = 𝐴𝐢 and 𝐴𝐷 = 𝐢𝐷.
How many of the following statements are true for the angles marked?
𝑀=π‘₯
π‘₯ + 𝑦 + 𝑧 = 180
𝑧 = 2π‘₯
A all of them
D none of them
B two
C one
E it depends on π‘₯
11. [JMC 2005 Q23] In the diagram, triangle π‘‹π‘Œπ‘ is isosceles, with π‘‹π‘Œ =
𝑋𝑍. What is the value of π‘Ÿ in terms of 𝑝 and π‘ž?
A
1
(𝑝
2
βˆ’ π‘ž)
D 𝑝+π‘ž
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B
1
(𝑝
2
+ π‘ž)
C π‘βˆ’π‘ž
E Impossible to determine
12. [JMC 2015 Q25] The four straight lines in the diagram are such that
π‘‰π‘ˆ = π‘‰π‘Š. The sizes of βˆ π‘ˆπ‘‹π‘, βˆ π‘‰π‘Œπ‘ and βˆ π‘‰π‘π‘‹ are π‘₯°, 𝑦° and 𝑧°.
Which of the following equations gives π‘₯ in terms of 𝑦 and 𝑧?
A π‘₯ =π‘¦βˆ’π‘§
B π‘₯ = 180 βˆ’ 𝑦 βˆ’ 𝑧
D π‘₯ = 𝑦 + 𝑧 βˆ’ 90
E π‘₯=
π‘¦βˆ’π‘§
2
𝑧
C π‘₯ =π‘¦βˆ’2
13. [JMO 2008 B3] In the diagram ABC and APQR are congruent triangles. The side PQ passes
through the point D and βˆ π‘ƒπ·π΄ = π‘₯°. Find an expression for βˆ π·π‘…π‘„ in terms of x.
14. [JMO 2008 A9] In the diagram, 𝐢𝐷 is the bisector of angle 𝐴𝐢𝐡.
Also 𝐡𝐢 = 𝐢𝐷 and 𝐴𝐡 = 𝐴𝐢. What is the size of angle 𝐢𝐷𝐴?
15. [JMO 2010 A10] Inn the diagram, 𝐽𝐾 and 𝑀𝐿 are parallel. 𝐽𝐾 = 𝐾𝑂 =
𝑂𝐽 = 𝑂𝑀 and 𝐿𝑀 = 𝐿𝑂 = 𝐿𝐾. Find the size of angle 𝐽𝑀𝑂.
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Exercise 5a – Simple Proofs
c. Points 𝐴, 𝐡 and 𝐢 lie in that
order on a straight line.
A point 𝐷, not on the line, is
placed such that
𝐴𝐷 = 𝐴𝐡, 𝐡𝐷 = 𝐡𝐢, and
∠𝐡𝐷𝐢 = 25°.
d. Points 𝑇 and π‘ˆ lie outside a
parallelogram 𝑃𝑄𝑅𝑆, and are
such that triangles 𝑅𝑄𝑇 and
π‘†π‘…π‘ˆ are equilateral and lie
wholly outside the
parallelogram.
Note that where relevant, full proofs are
required – a sequential list of justified
statements, not just a diagram with angles put
in.
1. a) Prove βˆ π‘…π‘„πΆ = 55°. Your proof
should only require one line and be in
the form β€œβˆ π‘Žπ‘›π‘”π‘™π‘’ =
π‘£π‘Žπ‘™π‘’π‘’ (π‘Ÿπ‘’π‘Žπ‘ π‘œπ‘›)”
3.
Prove the following:
a. ∠𝐡𝐢𝐴 = 20°
b) Prove that ∠𝐸𝐹𝐻 = 85°. (Your
proof should consist of two lines)
b.
∠𝐷𝐸𝐹 = 35°
c) Prove that ∠𝐡𝐢𝐷 = 2π‘₯. (Your proof
should consist of three lines)
Exercise 5b – More Complex Proofs
2. Draw diagrams which satisfy the
following criteria, ensuring you note
where lines are of equal length or
parallel. You do NOT need to find any
angles.
a. 𝐴𝐡𝐢 is a right-angled triangle
such that ∠𝐢𝐴𝐡 = 90°. 𝐷 is a
point on 𝐡𝐢 such that 𝐴𝐷 =
𝐡𝐷.
b. 𝐴𝐡𝐢 is an isosceles triangle
where 𝐴𝐡 = 𝐴𝐢. 𝐷 is a point
that lies inside the triangle
such that 𝐡𝐢𝐷 is equilateral.
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1. Prove the following.
a. 𝐴𝐷𝐢 and 𝐴𝐡𝐢 are rightangled triangles where
∠𝐴𝐢𝐷 = 10° and ∠𝐷𝐢𝐡 =
70°. Prove that triangle 𝐴𝐡𝐢
is isosceles.
b. 𝐴𝐢 = 𝐡𝐢, 𝐴𝐢𝐷 is a straight
line, ∠𝐡𝐢𝐴 = 40° and
∠𝐡𝐢𝐸 = 70° as per the
diagram. Prove that 𝐴𝐡 and
𝐢𝐸 are parallel.
2. As before, 𝐴𝐢 = 𝐡𝐢. Suppose also
that 𝐢𝐸 bisects the angle 𝐡𝐢𝐷, but
unlike before, no angles are known.
By introducing a suitable variable (you
may wish to start your proof β€œLet
βˆ π‘Žπ‘›π‘”π‘™π‘’ = π‘₯”) prove that 𝐴𝐡 and 𝐢𝐸
are parallel.
3. [JMO 2001 B3] In the diagram, B is the
midpoint of AC and the lines AP, BQ
and CR are parallel. The bisector of
βˆ π‘ƒπ΄π΅ meets BQ at Z.
Draw a diagram to show this, and join
Z to C.
(i) Given that βˆ π‘ƒπ΄π‘ = π‘₯°, find
βˆ π‘π΅πΆ in terms of x.
(ii) Show that CZ bisects βˆ π΅πΆπ‘….
4. [JMO 2015 B2] The diagram shows
triangle 𝐴𝐡𝐢, in which ∠𝐴𝐡𝐢 = 72°
and ∠𝐢𝐴𝐡 = 84°. The point 𝐸 lies on
𝐴𝐡 so that 𝐸𝐢 bisects ∠𝐡𝐢𝐴. The
point 𝐹 lies on 𝐢𝐴 extended. The
point 𝐷 lies on 𝐢𝐡 extended so that
𝐷𝐴 bisects ∠𝐡𝐴𝐹.
Prove that 𝐴𝐷 = 𝐢𝐸.
5. [JMO 2011 B4] In a triangle ABC, M
lies on AC and N lies on AB so that
βˆ π΅π‘πΆ = 4π‘₯°, βˆ π΅πΆπ‘ = 6π‘₯° and
βˆ π΅π‘€πΆ = βˆ πΆπ΅π‘€ = 5π‘₯°. Prove that
triangle ABC is isosceles.
6. [JMO 2015 B4] The point 𝐹 lies inside
the regular pentagon 𝐴𝐡𝐢𝐷𝐸 so that
𝐴𝐡𝐹𝐸 is a rhombus. Prove that 𝐸𝐹𝐢
is a straight line. (Hint: the interior
angle of a pentagon is 108°)
7. [Hamilton 2006 Q2] In triangle ABC,
∠𝐴𝐡𝐢 is a right angle. Points P and Q
lie on AC; BP is perpendicular to AC;
BQ bisects βˆ π΄π΅π‘ƒ. Prove that CB = CQ.
8. 𝐴𝐡𝐢 is a triangle such that ∠𝐴𝐡𝐢 =
90° and 𝐷 is a point on the line 𝐴𝐢
such that 𝐡𝐷 = 𝐢𝐷. By letting
∠𝐷𝐢𝐡 = π‘₯ or otherwise, prove that
𝐢𝐷 = 𝐷𝐴.
(hint: prove that Δ𝐴𝐡𝐷 is isosceles first).
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Solutions – Exercise 1
1. π‘Ž = 64°, 𝑏 = 70°, 𝑐 = 60°,
𝑑 = 112°, 𝑒 = 35°
2. 28
3. 50
4. 360
5. 104
6. 150
7. 170
8. 60
9. 130
10. Solution:135
11. Solution:1440
12. 150
13. 15
14. 40
15. 11
16. 240
Exercise 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
–
–
120°
–
71°
55°
75°
86°
27
360°
210
270
Suppose we add a line horizontal with
𝑅 as shown, where 𝑏 is split into 𝑏1
and 𝑏2 .
π‘Ž and 𝑏1 are cointerior thus π‘Ž + 𝑏1 =
180°.
𝑏2 and 𝑐 are cointerior thus 𝑏2 + 𝑐 =
180°
Thus
π‘Ž+𝑏+𝑐
= π‘Ž + 𝑏1 + 𝑏2 + 𝑐
= 180° + 180° = 360°
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Solutions – Exercise 3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
π‘₯ = 46°, 𝑦 = 74°, 𝑧 = 20°
30
140
40°
108°
35
55°
20°
4
65°
35°
90°
12°
21°
See slides for solutions to Exercises 4 and 5.
Angle Cheatsheet
Possible angles justifications:
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β€œAngles on a straight line sum to 180°.”
β€œAngles in a triangle sum to 180°.”
β€œAngles around a point sum to 360°.”
β€œBase angles of an isosceles triangle are equal.
β€œAlternate angles are equal.”
β€œCorresponding angles are equal.”
β€œVertically opposite angles are equal.”
β€œCointerior angles sum to 180°.”
β€œExterior angle of a triangle is the sum of the two other interior angles.”
Proof Format
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If you introduced any variables, start for example with: β€œLet ∠𝐴𝐡𝐢 = π‘₯.”
For statements about angles, it may be helpful to write in the form:
β€œβˆ π‘Žπ‘›π‘”π‘™π‘’ = π‘£π‘Žπ‘™π‘’π‘’ (π‘Ÿπ‘’π‘Žπ‘ π‘œπ‘›)”
where your reason is in the list above.
Where necessary have an appropriate conclusion:
Type of Proof
What you need to show
Some triangle is isosceles
Either show two angles are equal
or two sides are equal.
Show that the angles at 𝐡 sum to
180°, i.e. ∠𝐴𝐡𝐢 = 180°.
𝐴𝐡𝐢 is a straight line.
𝐴𝐡 = 𝐡𝐢
Two lines are parallel.
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We can show lengths are equal if
angles are equal, e.g. the triangle
is isosceles.
Show either the corresponding or
alternate angles on a connecting
line are equal.
How you might conclude your
proof
β€œβˆ π΄π΅πΆ = ∠𝐡𝐴𝐢 therefore Δ𝐴𝐡𝐢
is isosceles.”
β€œβˆ π΄π΅π· + ∠𝐷𝐡𝐢 = 180° therefore
𝐴𝐡𝐢 is a straight line.”
β€œβˆ π΄π΅πΆ = ∠𝐴𝐢𝐡 and therefore
Δ𝐴𝐡𝐢 is isosceles. Thus 𝐴𝐡 = 𝐴𝐢.”
(With respect to diagram on left)
β€œβˆ π΅π΄πΆ = ∠𝐷𝐢𝐸 and 𝐴𝐢𝐸 is a
straight line. Therefore 𝐴𝐡 and 𝐷𝐸
are parallel.”