Math  /  Geometry

QuestionLesson 13 Angles in Triangles Triangle Sum Theorem a b cc The sum of the three interior angles in a triangle is always 180180^{\circ}. a+b+c=180\angle a+\angle b+\angle c=180^{\circ} a
Find xx : \square Click to add text
Find the missing angles: <1= Click to add text <2= Click to add text <3= Click to add text <4= Click to add text <5= Click to add text <6= Click to add text \begin{array}{l} <1=\text { Click to add text } \\ <2=\text { Click to add text } \\ <3=\text { Click to add text } \\ <4=\text { Click to add text } \\ <5=\text { Click to add text } \\ <6=\text { Click to add text } \end{array}

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx in a triangle and then find the measures of a bunch of other angles formed by intersecting lines. Watch out! Don't forget that angles on a straight line add up to 180180^\circ and angles in a triangle *also* add up to 180180^\circ!

STEP 2

1. Find xx
2. Find angle 1
3. Find angle 2
4. Find angle 3
5. Find angle 4
6. Find angle 5
7. Find angle 6

STEP 3

Alright, so we know the angles in a triangle add up to 180180^\circ.
We've got a triangle with angles 100100^\circ, 5151^\circ, and (2x+3)(2x + 3)^\circ.
Let's add 'em up!
So, we can write the equation: 100+51+(2x+3)=180100 + 51 + (2x + 3) = 180

STEP 4

Let's combine the numbers on the left side: 100+51+3=154100 + 51 + 3 = \textbf{154}.
Now our equation looks like this: 154+2x=180154 + 2x = 180 To get 2x2x by itself, we subtract 154\textbf{154} from *both* sides: 154+2x154=180154154 + 2x - 154 = 180 - 154 2x=262x = \textbf{26}Now, we **divide both sides by 2** to find xx: 2x2=262\frac{2x}{2} = \frac{26}{2} x=13x = \textbf{13}Woohoo! We found xx!

STEP 5

We see that angle 1 and the 6868^\circ angle form a straight line, so they add up to 180180^\circ.

STEP 6

1+68=180 \angle 1 + 68 = 180 Subtract 6868 from both sides: 1+6868=18068 \angle 1 + 68 - 68 = 180 - 68 1=112 \angle 1 = \textbf{112}^\circ

STEP 7

Angle 2 and the 6868^\circ angle are **vertical angles**, which means they're equal!

STEP 8

So, 2=68\angle 2 = \textbf{68}^\circ.

STEP 9

Angle 3 and angle 1 are vertical angles, so they're equal!

STEP 10

Since 1=112\angle 1 = 112^\circ, then 3=112\angle 3 = \textbf{112}^\circ.

STEP 11

We're given that one angle is 9090^\circ, and we know angle 4 and the 9090^\circ angle make a straight line.

STEP 12

4+90=180 \angle 4 + 90 = 180 Subtract 9090 from both sides: 4+9090=18090 \angle 4 + 90 - 90 = 180 - 90 4=90 \angle 4 = \textbf{90}^\circ

STEP 13

Angle 5 and the given 9090^\circ angle are vertical angles!

STEP 14

Therefore, 5=90\angle 5 = \textbf{90}^\circ.

STEP 15

Angle 6 and angle 4 are vertical angles.

STEP 16

Since 4=90\angle 4 = 90^\circ, then 6=90\angle 6 = \textbf{90}^\circ.

STEP 17

x=13x = 13, 1=112\angle 1 = 112^\circ, 2=68\angle 2 = 68^\circ, 3=112\angle 3 = 112^\circ, 4=90\angle 4 = 90^\circ, 5=90\angle 5 = 90^\circ, and 6=90\angle 6 = 90^\circ.

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