Expressions, Equations, and Relationships>Writing and Solving Equations Using Geometry Concepts and Angle Relationships(TEKS.Math.7.11.C)
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Texas 7th Grade Math › Expressions, Equations, and Relationships>Writing and Solving Equations Using Geometry Concepts and Angle Relationships(TEKS.Math.7.11.C)
The angles in a triangle measure $x^\circ$, $(2x + 15)^\circ$, and $(x - 5)^\circ$. Which equation can you write using the triangle angle-sum relationship?
$x + (2x + 15) = (x - 5)$
$x + (2x + 15) + (x - 5) = 180$
$x + (2x + 15) + (x - 5) = 90$
$(2x + 15) = (x - 5)$
Explanation
Triangle angle sum: $180^\circ$. Set up $x + (2x + 15) + (x - 5) = 180$. Simplify: $4x + 10 = 180 \Rightarrow 4x = 170 \Rightarrow x = 42.5$. The angles are $42.5^\circ$, $100^\circ$, and $37.5^\circ$, which add to $180^\circ$. Distractors either set angles equal or use $90^\circ$ incorrectly.
Two supplementary angles measure $(3y + 20)^\circ$ and $(2y - 10)^\circ$. What is the measure of each angle?
110° and 70°
95° and 85°
128° and 52°
122° and 58°
Explanation
Supplementary angles sum to $180^\circ$. Set up $(3y + 20) + (2y - 10) = 180$. Then $5y + 10 = 180 \Rightarrow 5y = 170 \Rightarrow y = 34$. Angles: $3(34)+20 = 122^\circ$ and $2(34)-10 = 58^\circ$. Check: $122^\circ + 58^\circ = 180^\circ$.
Two vertical angles measure $(x + 25)^\circ$ and $(3x - 35)^\circ$. What is the value of $x$?
30
20
45
60
Explanation
Vertical angles are congruent, so set $x + 25 = 3x - 35$. Solve: $25 + 35 = 3x - x \Rightarrow 60 = 2x \Rightarrow x = 30$. Check: $(x+25)^\circ = 55^\circ$ and $(3x-35)^\circ = 55^\circ$, confirming they are equal.
In a triangle, the exterior angle at one vertex measures $(4z + 10)^\circ$. The two remote interior angles measure $(z + 25)^\circ$ and $(2z + 5)^\circ$. What is the value of $z$?
10
15
20
30
Explanation
Exterior angle theorem: an exterior angle equals the sum of the two remote interior angles. Set $4z + 10 = (z + 25) + (2z + 5)$. Then $4z + 10 = 3z + 30 \Rightarrow z = 20$. Check: remote interiors $45^\circ$ and $45^\circ$ sum to $90^\circ$, and the exterior angle $4(20)+10 = 90^\circ$.
A right triangle has one acute angle $(3x - 5)^\circ$ and the other acute angle $(x + 15)^\circ$. What are the measures of the two acute angles?
60° and 30°
55° and 35°
70° and 20°
50° and 40°
Explanation
In a right triangle, the two acute angles are complementary, so they sum to $90^\circ$. Set $(3x - 5) + (x + 15) = 90 \Rightarrow 4x + 10 = 90 \Rightarrow 4x = 80 \Rightarrow x = 20$. The angles are $3(20)-5 = 55^\circ$ and $20+15 = 35^\circ$. Check: $55^\circ + 35^\circ = 90^\circ$.