Download 2.3: Angle Properties in Triangles Can you prove that the

2.3: Angle Properties in Triangles
Can you prove that the sum of the measures of the interior angles of any
triangle is 180o?
A.
Draw an acute triangle, ∆RED. Construct line PQ through vertex D, parallel to RE.
B.
Identify pairs of equal angles in your diagram. Explain how you know that the
measures of the angles in each pair are equal.

C.
What is the sum of the measures of ∠PDR, ∠RDE, and ∠QDE? Explain how you know.

D.
PDR =DRE; QDE = RED Alternate interior angles
180°, because the three angles form a straight line
Explain why ∠𝑫𝑹𝑬 + ∠𝑹𝑫𝑬 + ∠𝑹𝑬𝑫 = 𝟏𝟖𝟎𝒐 . (prove)
1.
2.
3.
4.
E.
Statement
PDR +∠RDE +∠QDE = 180°
∠DRE = ∠PDR
∠RED = ∠QDE
DRE +∠RDE +∠RED = 180°
Reason
These angles form a straight line
Alternate interior angles
Alternate interior angles
Substitute angles in step 2 & 3 in equation in
step 1.
Repeat parts A to D, first for an obtuse triangle and then for a right triangle. Are
your results the same as they were for the acute triangle?
Example 1 – In the diagram, ∠MTH is an exterior angle of
∆MAT. Determine the measures of the unknown
angles in ∆MAT.
MTA + MTH = 180o
MTA + 155o = 180o
MTA = 180o – 155o = 25o
TAM + AMT +  MTA = 180o
TAM + 40o + 25o = 180o
TAM = 180o – 40o -25o = 115o

If you are given one interior angle and one exterior angle of a triangle, can you always
determine the other interior angles of the triangle? Explain, using diagrams.
If the interior angle you are given is adjacent to a known
exterior angle, then you cannot determine the other angles. For
example, in ∆PQR, neither of the non-adjacent interior angles
are known, so there is not enough information to determine the
unknown interior angles.
Example 2 – Determine the relationship between an exterior angle of a triangle and its
non-adjacent interior angles.
d +  c = 180o  on the same line
c + a + b = 180o  angles in a triangle add to
180o
 d + c = c + a + b  since the 2 equations
above both equal 180o they must also equal each
other
d = a + b  get d by itself
So an exterior angle is equal to the sum of the two non-adjacent interior angles of a
triangle.
Example 3 –
(a) Determine the measures of ∠NMO, ∠MNO, ∠QMO.
LMN = MNP = 67o
MNP = MNO + PNO (Alt int)
67o = MNO + 20o
MNO = 67o – 20o = 47o
NMO +  MON +  ONM = 180o (SATS)
NMO + 39o + 47o = 180o
NMO = 180o – 39o – 47o = 94o
MNP + NMQ = 180o (int. ’s on same side of trans. suppl.)
67o + NMQ = 180o
NMQ = 180o – 67o =113o
NMQ =  NMO +  OMQ
113o = 94o + OMQ
OMQ = 113o – 94o = 19o
(b) If QP∥MR, determine the measures of ∠MQO, ∠MOQ, ∠NOP, ∠OPN, and ∠RNP.
LMN = MQO (Corr. ’s) = 67o
MOQ + OQM + QMO = 180o
MOQ + 67o + 19o = 180o
MOQ = 180o – 67o – 19o =94o
MNP + OPN = 180o (int ’s same side)
67o + OPN = 180o
OPN = 180o – 67o = 113o
OPN + PNO +  NOP = 180o (SATS)
113o + 20o + NOP = 180o
NOP = 180o – 113o – 20o = 47o
RNP = NPO (Alt Int) = 113o
Example 4: Prove: A = 30°v
∆BCD is an equilateral triangle which means all 3 sides are the
same and all 3 angles are the same. If all of the angles need to add
to 180o and all of the angles are the same, the angles must each
equal 180o  3
So, CBD = BCD = CDB = 180o  3= 60o
∆ADB is an isosceles triangle. This means that 2 sides are equal (AD = BD) and 2 angles are
equal (DBA =  BAD).
BDA = 180o - CDB = 180o – 60o = 120o
This means the other two angles in ∆ADB need to add to 60o (because all three need to add to
180o and one of the angles is known to be 120o). We also know that the two angles we have left
need to have the same measure (since this is an isosceles triangle).
DBA = BAD = 60o  2 = 30o
Therefore: A = 30o
Key Ideas:

You can prove properties of angles in triangles using other properties that have already
been proven.

In any triangle, the sum of the measures of the interior angles is proven to be 180o.

The measure of any exterior angle of a triangle is proven to be equal to the sum of the
measures of the two non-adjacent interior angles.
2.3 Assignment: Nelson Foundations of Mathematics 11, Sec 2.3, pg. 90-93
Questions: 2, 3, 6, 7, 9, 12a, 14