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3 years ago

11

Lines m and n are parallel, as shown in the diagram below. What are the measures of angles A and B? Hint: The sum of all interio

r angles of a triangle must equal 180 degrees.

2 answers:

sveticcg [70]3 years ago

8 0

Answer:

A = 55

B = 60

Step-by-step explanation:

We know that 55+ b+ other angle = 180 since they make a straight line

The other angle = 65 since they are alternate interior angles

55+ B+ 65 = 180

Combine like terms

120 + B = 180

B = 60

A + B + 65 = 180 interior angles of a triangle must equal 180 degrees

A +60+ 65 =180

Combine like terms

A +125 = 180

A = 55

mel-nik [20]3 years ago

4 0

Answer:

$\boxed{A = 55\°}$

$\boxed{B = 60\°}$

Step-by-step explanation:

Exterior Angle with A = 180 - 55 = 125 degrees (Angles on a straight line)

The measure of exterior angle is equal to the sum of non-adjacent interior angles.

So,

125° = B + 65°

B = 125 - 65

B = 60°

Now,

A = 180 - 60 - 65 (Interior angles of a triangle add up to 180 degrees)

A = 55°

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3 years ago

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3 years ago

Read 2 more answers

If <img src="https://tex.z-dn.net/?f=x%20%3D%209%20-%204%5Csqrt%7B5%7D" id="TexFormula1" title="x = 9 - 4\sqrt{5}" alt="x = 9 -

Komok [63]

Observe that

$\left(\sqrt x - \dfrac1{\sqrt x}\right)^2 = \left(\sqrt x\right)^2 - 2\sqrt x\dfrac1{\sqrt x} + \left(\dfrac1{\sqrt x}\right)^2 = x - 2 + \dfrac1x$

Now,

$x = 9 - 4\sqrt5 \implies \dfrac1x = \dfrac1{9-4\sqrt5} = \dfrac{9 + 4\sqrt5}{9^2 - \left(4\sqrt5\right)^2} = 9 + 4\sqrt5$

so that

$\left(\sqrt x - \dfrac1{\sqrt x}\right)^2 = (9 - 4\sqrt5) - 2 + (9 + 4\sqrt5) = 16$

$\implies \sqrt x - \dfrac1{\sqrt x} = \pm\sqrt{16} = \pm 4$

To decide which is the correct value, we need to examine the sign of $\sqrt x - \frac1{\sqrt x}$ . It evaluates to 0 if

$\sqrt x = \dfrac1{\sqrt x} \implies x = 1$

We have

$9 - 4\sqrt5 = \sqrt{81} - \sqrt{16\cdot5} = \sqrt{81} - \sqrt{80} > 0$

Also,

$\sqrt{81} - \sqrt{64} = 9 - 8 = 1$

and $\sqrt x$ increases as $x$ increases, which means

$0 < 9 - 4\sqrt5 < 1$

Therefore for all $0 < x < 1$ ,

$\sqrt x - \dfrac1{\sqrt x} < 0$

For example, when $x=\frac14$ , we get

$\sqrt{\dfrac14} - \dfrac1{\sqrt{\frac14}} = \dfrac1{\sqrt4} - \sqrt4 = \dfrac12 - 2 = -\dfrac32 < 0$

Then the target expression has a negative sign at the given value of $x$ :

$x = 9-4\sqrt5 \implies \sqrt x - \dfrac1{\sqrt x} = \boxed{-4}$

Alternatively, we can try simplifying $\sqrt x$ by denesting the radical. Let $a,b,c$ be non-zero integers ( $c>0$ ) such that

$\sqrt{9 - 4\sqrt5} = a + b\sqrt c$

Note that the left side must be positive.

Taking squares on both sides gives

$9 - 4\sqrt5 = a^2 + 2ab\sqrt c + b^2c$

Let $c=5$ and $ab=-2$ . Then

$a^2+5b^2=9 \implies a^2 + 5\left(-\dfrac2a\right)^2 = 9 \\\\ \implies a^2 + \dfrac{20}{a^2} = 9 \\\\ \implies a^4 + 20 = 9a^2 \\\\ \implies a^4 - 9a^2 + 20 = 0 \\\\ \implies (a^2 - 4) (a^2 - 5) = 0 \\\\ \implies a^2 = 4 \text{ or } a^2 = 5$

$a^2 = 4 \implies 5b^2 = 5 \implies b^2 = 1$

$a^2 = 5 \implies 5b^2 = 4 \implies b^2 = \dfrac45$

Only the first case leads to integer coefficients. Since $ab=-2$ , one of $a$ or $b$ must be negative. We have

$a^2 = 4 \implies a = 2 \text{ or } a = -2$

Now if $a=2$ , then $b=-1$ , and

$\sqrt{9 - 4\sqrt5} = 2 - \sqrt5$

However, $\sqrt5 > \sqrt4 = 2$ , so $2-\sqrt5$ is negative, so we don't want this.

Instead, if $a=-2$ , then $b=1$ , and thus

$\sqrt{9 - 4\sqrt5} = -2 + \sqrt5$

Then our target expression evaluates to

$\sqrt x - \dfrac1{\sqrt x} = -2 + \sqrt5 - \dfrac1{-2 + \sqrt5} \\\\ ~~~~~~~~~~~~ = -2 + \sqrt5 - \dfrac{-2 - \sqrt5}{(-2)^2 - \left(\sqrt5\right)^2} \\\\ ~~~~~~~~~~~~ = -2 + \sqrt5 + \dfrac{2 + \sqrt5}{4 - 5} \\\\ ~~~~~~~~~~~~ = -2 + \sqrt5 - (2 + \sqrt5) = \boxed{-4}$

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2 years ago

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3 years ago

The shorter leg of a right triangle is 21 feet less than the other leg. Find the length of the two legs if the hypotenuse is 39

mina [271]

Answer:

leg 1= 36 feet

leg2 = 15 feet

Step-by-step explanation:

Hi, we have to apply the Pythagorean Theorem:

c^2 = a^2 + b^2

Where c is the hypotenuse of the triangle (in this case the distance between Doreen’s house and the tower) and a and b are the other legs.

leg1 = x

leg2 = x-21 (21 feet less than the other leg)

Replacing with the values given:

39^2 = x^2 + (x-21)^2

1,521 = x^2 + x^2 -42x +441

0 = 2x^2 -42x +441-1,521

0= 2x^2 -42x -1,080

For: ax2+ bx + c

x =[ -b ± √b²-4ac] /2a (quadratic formula)

Replacing with the values given:

x=-(-42)± √(-42)²-4(2)-1080] /2(2)

x= 42± √10,404] /4

x = 42± 102 /4

Positive:

x = 42+102 /4 = 36

leg 1= 36

leg2 = 36-21 =15

Feel free to ask for more if needed or if you did not understand something.

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3 years ago