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Unit 8: Geometry and Trigonometry

Topic 2

Triangle Fundamentals

Every triangle has three interior angles whose measures add up to exactly 180∘180^\circ. This is the Triangle Angle Sum Theorem, and it is the single most-used geometry fact on the SAT. If you know two angles in a triangle, you can always find the third by subtracting from 180∘180^\circ. Many SAT problems give you two angles (or expressions for two angles) and ask for the third, so practice setting up and solving the equation quickly.
An exterior angle of a triangle is formed when one side of the triangle is extended. The Exterior Angle Theorem says that the measure of an exterior angle equals the sum of the two non-adjacent interior angles (sometimes called the two ``remote interior angles''). This shortcut saves a step compared to finding the third interior angle first and then subtracting from 180∘180^\circ. For instance, if two interior angles of a triangle are 40∘40^\circ and 65∘65^\circ, the exterior angle adjacent to the third angle is 40∘+65∘=105∘40^\circ + 65^\circ = 105^\circ.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. In other words, for sides aa, bb, and cc:
a+b>c,a+c>b,b+c>a.a + b > c, \quad a + c > b, \quad b + c > a.
This means if you are given two sides of a triangle, the third side must be longer than the absolute difference and shorter than the sum of the other two. For example, if two sides are 5 and 9, the third side must satisfy 9−5<c<9+59 - 5 < c < 9 + 5, i.e.\ 4<c<144 < c < 14.
One more useful relationship: the longest side of a triangle is always opposite the largest angle, and the shortest side is opposite the smallest angle. If two sides are equal, the angles opposite them are also equal (the triangle is isosceles). If all three sides are equal, all three angles are 60∘60^\circ (the triangle is equilateral).

Core Skills

  • Use the Triangle Angle Sum Theorem to find a missing angle when two angles (or algebraic expressions for angles) are known.
  • Apply the Exterior Angle Theorem to find an exterior angle or to set up an equation involving remote interior angles.
  • Determine whether three given side lengths can form a triangle using the Triangle Inequality Theorem.
  • Find the range of possible lengths for a missing side of a triangle.
  • Relate side lengths to opposite angle measures to compare or order sides and angles.

Example 1: Finding a Missing Angle

In triangle PQRPQR, ∠P=52∘\angle P = 52^\circ and ∠Q=73∘\angle Q = 73^\circ. What is the measure of ∠R\angle R? Step 1: Apply the Triangle Angle Sum Theorem.
∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^\circ
Step 2: Substitute and solve.
52+73+∠R=180  ⟹  ∠R=180−125=55∘52 + 73 + \angle R = 180 \implies \angle R = 180 - 125 = 55^\circ
The measure of ∠R\angle R is 55\boxed{55} degrees.

Example 2: Algebraic Angle Expressions

The three angles of a triangle measure (2x+10)∘(2x + 10)^\circ, (3x−5)∘(3x - 5)^\circ, and (x+25)∘(x + 25)^\circ. Find the value of xx. Step 1: Set the sum equal to 180∘180^\circ.
(2x+10)+(3x−5)+(x+25)=180(2x + 10) + (3x - 5) + (x + 25) = 180
Step 2: Combine like terms.
6x+30=1806x + 30 = 180
Step 3: Solve for xx.
6x=150  ⟹  x=256x = 150 \implies x = 25
The value of xx is 25\boxed{25}.

Example 3: Exterior Angle Theorem

In a triangle, the two non-adjacent interior angles measure 48∘48^\circ and 57∘57^\circ. What is the measure of the exterior angle adjacent to the third interior angle? Step 1: Apply the Exterior Angle Theorem directly.
Exterior angle=48+57=105∘\text{Exterior angle} = 48 + 57 = 105^\circ
The exterior angle measures 105\boxed{105} degrees.

Example 4: Algebraic Exterior Angle

In triangle ABCABC, ∠A=(3x)∘\angle A = (3x)^\circ and ∠B=(x+20)∘\angle B = (x + 20)^\circ. An exterior angle at vertex CC measures (5x−10)∘(5x - 10)^\circ. Find the value of xx. Step 1: By the Exterior Angle Theorem, the exterior angle at CC equals ∠A+∠B\angle A + \angle B.
5x−10=3x+(x+20)5x - 10 = 3x + (x + 20)
Step 2: Simplify and solve.
5x−10=4x+20  ⟹  x=305x - 10 = 4x + 20 \implies x = 30
The value of xx is 30\boxed{30}.

Example 5: Triangle Inequality

Two sides of a triangle have lengths 7 and 11. If the third side has integer length, how many possible values are there for the third side? Step 1: Apply the Triangle Inequality. The third side cc must satisfy:
11−7<c<11+7  ⟹  4<c<1811 - 7 < c < 11 + 7 \implies 4 < c < 18
Step 2: Count the integers in the range 4<c<184 < c < 18.
c∈{5,6,7,8,9,10,11,12,13,14,15,16,17}c \in \{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}
There are 13\boxed{13} possible integer values.

Example 6: Side-Angle Relationship

In triangle DEFDEF, DE=8DE = 8, EF=12EF = 12, and DF=10DF = 10. List the angles of the triangle in order from smallest to largest. Step 1: Identify the shortest and longest sides. The shortest side is DE=8DE = 8 and the longest side is EF=12EF = 12. Step 2: The smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.
  • DE=8DE = 8 is opposite ∠F\angle F (smallest angle)
  • DF=10DF = 10 is opposite ∠E\angle E (middle angle)
  • EF=12EF = 12 is opposite ∠D\angle D (largest angle)
From smallest to largest: ∠F,∠E,∠D\boxed{\angle F, \angle E, \angle D}.

Key Takeaways

  • The three interior angles of any triangle sum to 180∘180^\circ. Use this to find missing angles, especially when angles are given as algebraic expressions.
  • An exterior angle of a triangle equals the sum of the two remote interior angles. This often saves a step over finding the third interior angle first.
  • For the Triangle Inequality, remember the range for the third side: it must be strictly between the difference and the sum of the other two sides.
  • The largest angle is always opposite the longest side. This relationship is frequently tested when the SAT asks you to compare or order angles.