Math  /  Geometry

QuestionList the angles and sides of each triangle in order from smallest to largest. 22. 23.

Studdy Solution

STEP 1

What is this asking? We need to figure out the size of all the angles and then list the sides and angles of each triangle from smallest to largest. Watch out! Remember, the smallest angle is always opposite the shortest side, and the largest angle is always opposite the longest side!

STEP 2

1. Solve Triangle 22
2. Solve Triangle 23

STEP 3

Alright, let's **start** with triangle 22!
We've got a right angle, which means one angle is 90\mathbf{90^\circ}.
We also know the other two angles: (2x+1) (2x + 1)^\circ and (2x+9) (2x + 9)^\circ .

STEP 4

Since the angles in *any* triangle add up to 180 180^\circ , we can **write** an equation: (2x+1)+(2x+9)+90=180 (2x + 1) + (2x + 9) + 90 = 180

STEP 5

Let's **simplify** and **solve** for x x : 4x+100=180 4x + 100 = 180 4x=180100 4x = 180 - 100 4x=80 4x = 80 x=804 x = \frac{80}{4} x=20 x = \mathbf{20} Woohoo! We found x x !

STEP 6

Now, let's **plug** x=20\mathbf{x = 20} back into the expressions for the angles: 2x+1=2(20)+1=41 2x + 1 = 2(20) + 1 = 41 2x+9=2(20)+9=49 2x + 9 = 2(20) + 9 = 49 So, our angles are 41\mathbf{41^\circ}, 49\mathbf{49^\circ}, and 90\mathbf{90^\circ}.

STEP 7

Finally, let's **list** the sides and angles in order from smallest to largest.
Remember, the smallest angle is opposite the shortest side!
Let's say the sides opposite the 4141^\circ, 4949^\circ, and 9090^\circ angles are aa, bb, and cc, respectively.
Then we have: Angles: 41 41^\circ , 49 49^\circ , 90 90^\circ Sides: a a , b b , c c

STEP 8

Okay, onto triangle 23!
We know all three angles in terms of x x : (x1) (x - 1)^\circ , (x+6) (x + 6)^\circ , and (2x+3) (2x + 3)^\circ .

STEP 9

Again, the angles in a triangle add up to 180 180^\circ , so we can **write** another equation: (x1)+(x+6)+(2x+3)=180 (x - 1) + (x + 6) + (2x + 3) = 180

STEP 10

Let's **simplify** and **solve** for x x : 4x+8=180 4x + 8 = 180 4x=1808 4x = 180 - 8 4x=172 4x = 172 x=1724 x = \frac{172}{4} x=43 x = \mathbf{43} Yes! We got x x !

STEP 11

Now, let's **substitute** x=43\mathbf{x = 43} back into the angle expressions: x1=431=42 x - 1 = 43 - 1 = 42 x+6=43+6=49 x + 6 = 43 + 6 = 49 2x+3=2(43)+3=89 2x + 3 = 2(43) + 3 = 89 So, our angles are 42\mathbf{42^\circ}, 49\mathbf{49^\circ}, and 89\mathbf{89^\circ}.

STEP 12

Let's say the sides opposite the 4242^\circ, 4949^\circ, and 8989^\circ angles are dd, ee, and ff, respectively.
Then, **listing** the sides and angles from smallest to largest, we get: Angles: 42 42^\circ , 49 49^\circ , 89 89^\circ Sides: d d , e e , f f

STEP 13

For triangle 22, the angles are 41 41^\circ , 49 49^\circ , and 90 90^\circ , and the sides opposite those angles are a a , b b , and c c , respectively, ordered from smallest to largest.
For triangle 23, the angles are 42 42^\circ , 49 49^\circ , and 89 89^\circ , and the sides opposite those angles are d d , e e , and f f , respectively, ordered from smallest to largest.

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