AMC 8 · 2009 · #19
Grade 8 geometry-2dProblem
Two angles of an isosceles triangle measure and . What is the sum of the three possible values of ?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: An isosceles triangle has two of its angles measuring $70^\circ$ and $x^\circ$. Because we are not told which role each angle plays (base angle or vertex angle), there is more than one possible value of $x$. Find every possible $x$ and add them up.
Givens: The triangle is isosceles, so two of its three angles are equal; Two of the angles measure $70^\circ$ and $x^\circ$; Answer choices: (A) $95$, (B) $125$, (C) $140$, (D) $165$, (E) $180$
Unknowns: All possible values of $x$; The sum of those possible values
Understand
Restated: An isosceles triangle has two of its angles measuring $70^\circ$ and $x^\circ$. Because we are not told which role each angle plays (base angle or vertex angle), there is more than one possible value of $x$. Find every possible $x$ and add them up.
Givens: The triangle is isosceles, so two of its three angles are equal; Two of the angles measure $70^\circ$ and $x^\circ$; Answer choices: (A) $95$, (B) $125$, (C) $140$, (D) $165$, (E) $180$
Plan
Primary tool: #2 Make a Systematic List
Secondary: #1 Draw a Diagram
The phrase "sum of the three possible values" tells us there are exactly three cases — a perfect cue for Tool #2 (Systematic List). We organize the cases by asking which two of the three angles are the equal pair: either the two $70^\circ$ angles are equal, the two $x^\circ$ angles are equal, or the $70^\circ$ and $x^\circ$ are the equal pair. Tool #1 (Draw a Diagram) supports this — a quick sketch with the equal sides marked makes each case concrete and prevents double-counting.
Execute — Answer: D
4.G.A.2 Step 1 - Sketch a generic isosceles triangle and label its three angles.
- Two of them must be equal (the "base angles"), and the third is the "vertex angle." Our job is to decide which slot each given angle ($70^\circ$ or $x^\circ$) fills.
💡 Recognizing the isosceles structure — two equal angles plus one other — is the Grade 4 "classify triangles by their properties" skill.
8.G.A.5 Step 2 - Case 1: the two equal angles are both $70^\circ$, and $x^\circ$ is the third (vertex) angle.
- Use the fact that the three angles add to $180^\circ$.
💡 The triangle-angle-sum fact ($180^\circ$) is the Grade 8 "informal arguments about triangle angles" standard.
8.G.A.5 Step 3 Case 2: the two equal angles are both $x^\circ$, and $70^\circ$ is the third (vertex) angle.
💡 Same angle-sum equation, different slot for the unknown — a clean Tool #2 sub-case.
4.G.A.2 Step 4 - Case 3: one $70^\circ$ angle is paired with the $x^\circ$ angle as the two equal angles.
- Equal angles must have equal measures, so $x = 70$.
- (The third angle is then $180 - 70 - 70 = 40^\circ$, which is positive — a valid triangle.)
💡 By definition, the two base angles of an isosceles triangle have the same measure — this is the Grade 4 classification fact.
4.MD.C.7 Step 5 - All three cases give valid triangles (each third angle is positive), so the three possible values of $x$ are $40$, $55$, and $70$.
- Add them.
💡 Adding the three possible angle measures uses the Grade 4 "angles are additive" skill in its simplest form.
4.G.A.2 Sketch a generic isosceles triangle and label its three angles. Two of them must 8.G.A.5 Case 1: the two equal angles are both $70^\circ$, and $x^\circ$ is the third (ve 8.G.A.5 Case 2: the two equal angles are both $x^\circ$, and $70^\circ$ is the third (ve 4.G.A.2 Case 3: one $70^\circ$ angle is paired with the $x^\circ$ angle as the two equal 4.MD.C.7 All three cases give valid triangles (each third angle is positive), so the thre Review
Reasonableness: Each of $x = 40, 55, 70$ produces a real triangle: $(70,70,40)$, $(55,55,70)$, $(70,70,40)$ — wait, Case 3's triangle is the same set of angles as Case 1, but $x$ refers to a different angle, so the value of $x$ ($70$ vs $40$) really is different. All three triangles satisfy the $180^\circ$ sum and have two equal angles, so each $x$ is legitimate. The total $40 + 55 + 70 = 165$ matches choice (D), and the answer is below the $180$ cap of choice (E), as it should be since no single $x$ can equal $180$.
Alternative: Tool #3 (Eliminate Possibilities) on the choices: the three values of $x$ must include $70$ (the case where $x$ is a base angle matching the given $70^\circ$), so the sum is at least $70$. Choices (A) $95$ and (B) $125$ are too small to also contain the $40$ and $55$ cases; (E) $180$ is too big (it would need a $0^\circ$ angle somewhere). Only (C) $140$ and (D) $165$ remain, and a one-line check ($40 + 55 + 70$) confirms (D).
CCSS standards used (min grade 8)
4.G.A.2Classify two-dimensional figures based on properties of their lines and angles (Identifying an isosceles triangle as having two equal base angles, which is the structural fact that creates the three cases.)4.MD.C.7Recognize angle measure as additive and find unknown angle measures (Adding the three possible angle values ($40 + 55 + 70$) to get the requested sum, and treating the triangle's angles as additive parts of $180^\circ$.)8.G.A.5Use informal arguments to establish facts about the angle sum of triangles (Applying the fact that the three angles of a triangle sum to $180^\circ$ to solve $70 + 70 + x = 180$ and $x + x + 70 = 180$.)
⭐ When a problem says "the possible values," make a short list of every case — here, just three ways to label the equal angles — and the answer falls out by adding.
⭐ When a problem says "the possible values," make a short list of every case — here, just three ways to label the equal angles — and the answer falls out by adding.
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