Triangle Angle Theorems - Math 1

1. Triangle $MNP$ has exterior angles measuring $110°$, $125°$, and $125°$ at vertices $M$, $N$, and $P$ respectively. What is the measure of the largest interior angle of the triangle?
   Answer: $70°$ Each exterior angle and its corresponding interior angle are supplementary (sum to $180°$). So the interior angles are: at $M$: $180° - 110° = 70°$, at $N$: $180° - 125° = 55°$, at $P$: $180° - 125° = 55°$. Let's verify: $70° + 55° + 55° = 180°$ ✓. The largest interior angle is $70°$. Choice (A) $55°$ gives one of the smaller interior angles. Choice (C) $75°$ and choice (D) $85°$ are incorrect calculations that don't correspond to any valid interior angle in this triangle.
2. In triangle $RST$, angle $R = (4y - 15)°$, angle $S = (2y + 25)°$, and angle $T = (y + 20)°$. An exterior angle at vertex $S$ measures $(5y + 10)°$. Which equation correctly represents the relationship needed to solve for $y$?
   Answer: $(4y - 15) + (y + 20) = (5y + 10)$ By the exterior angle theorem, an exterior angle equals the sum of the two non-adjacent interior angles. The exterior angle at vertex $S$ equals the sum of angles $R$ and $T$: $(4y - 15) + (y + 20) = (5y + 10)$. Choice (B) represents the triangle sum theorem, not the exterior angle theorem. Choice (C) incorrectly sets the interior angle at $S$ plus the exterior angle equal to $180°$, which is correct but doesn't use the exterior angle theorem. Choice (D) incorrectly includes angle $S$ (the adjacent interior angle) instead of using only the non-adjacent angles $R$ and $T$.
3. In triangle ABC, the measure of angle A is 20° more than twice the measure of angle B, and the measure of angle C is 10° less than three times the measure of angle B. What is the measure of angle B?
   Answer: 28° Let angle B measure x°. Then angle A measures (2x + 20)° and angle C measures (3x - 10)°. Using the triangle sum theorem: x + (2x + 20) + (3x - 10) = 180. Simplifying: 6x + 10 = 180, so 6x = 170, giving x = 170/6 = 85/3 ≈ 28.33°. Since the answer choices are whole numbers, we round to 28°. Check: angles are approximately 28°, 76°, and 74°, which sum to 178°, very close to 180° due to rounding.
4. In triangle $XYZ$, the measures of the three interior angles are in the ratio $2:3:4$. If an exterior angle is drawn at the vertex corresponding to the largest interior angle, what is the measure of this exterior angle?
   Answer: $100°$ Let the angles be $2x$, $3x$, and $4x$. Using the triangle sum theorem: $2x + 3x + 4x = 180°$, so $9x = 180°$ and $x = 20°$. The angles are $40°$, $60°$, and $80°$. The largest interior angle is $80°$. The exterior angle at this vertex equals the sum of the two non-adjacent interior angles: $40° + 60° = 100°$. Choice (A) $80°$ incorrectly gives the measure of the largest interior angle itself. Choice (C) $120°$ would result from incorrectly adding $60° + 80°$. Choice (D) $140°$ would result from incorrectly adding $40° + 80°$ and making an arithmetic error.
5. In triangle JKL, the ratio of the three interior angles is $2:3:4$. If an exterior angle is drawn at the vertex with the smallest interior angle, what is the measure of this exterior angle?
   Answer: $140°$ If the angles are in the ratio $2:3:4$, then they can be written as $2k$, $3k$, and $4k$ for some value $k$. Using the triangle sum theorem: $2k + 3k + 4k = 180°$, so $9k = 180°$ and $k = 20°$. The three angles are $40°$, $60°$, and $80°$. The smallest angle is $40°$, so the exterior angle at that vertex is $180° - 40° = 140°$.

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