Question

solve for x.
(first triangle: angles x + 59, x + 51, 84°)
options for first solve x: a) 63° b) -46 c) 46°
(second triangle: angles 70°, 65°, x + 53)
solve for x.
options: a) -10 b) -12 c) -8 d) -5

1. what kind of triangle is this? (triangle with angles 39°, 91°, 50°)

options: a) obtuse b) acute c) right

1. match the following triangles with the correct definition

options (for triangle definitions):

• all sides are the same length
• has at least 2 equal sides
• has no equal sides
• has one right angle
• has 3 acute angles

Explanation:

First, let's solve the triangle angle problem (the one with angles \(70^\circ\), \(65^\circ\), and \(x + 53\)):

Step 1: Recall the triangle angle sum theorem

The sum of the interior angles of a triangle is \(180^\circ\). So we set up the equation: \(70^\circ + 65^\circ + (x + 53) = 180^\circ\).

Step 2: Simplify the left - hand side

First, add \(70\) and \(65\): \(70+65 = 135\). So the equation becomes \(135+(x + 53)=180\). Then, combine like terms: \(x+135 + 53=x + 188\). So we have \(x + 188=180\).

Step 3: Solve for \(x\)

Subtract \(188\) from both sides of the equation: \(x=180 - 188=- 8\)? Wait, that can't be right. Wait, maybe the \(x + 53\) is an angle? Wait, no, maybe I misread. Wait, the triangle has angles \(70^\circ\), \(65^\circ\), and the third angle is \(x + 53\) (as an angle measure). Wait, \(70+65+(x + 53)=180\). Let's recalculate: \(70+65 = 135\), \(135+53=188\), so \(x+188 = 180\), \(x=180 - 188=-8\). But let's check the other triangle (the one with \(84^\circ\), \(x + 59\), \(x + 51\)):

For the triangle with angles \(84^\circ\), \(x + 59\), \(x + 51\):

Step 1: Apply the triangle angle sum theorem

\(84+(x + 59)+(x + 51)=180\)

Step 2: Simplify the left - hand side

Combine like terms: \(84+59 + 51+2x=84 + 110+2x=194+2x\)

Step 3: Solve for \(x\)

Set \(194+2x = 180\), then \(2x=180 - 194=-14\), \(x=-7\). But the options for the first triangle (with \(70\) and \(65\)) have options a) - 10, b) - 12, c) - 8, d) - 5. Wait, maybe the \(x + 53\) is a typo or I misread. Wait, maybe the triangle has angles \(70^\circ\), \(65^\circ\), and the third angle is \(x+53\) (in degrees), and we made a mistake. Wait, \(70 + 65=135\), \(180-135 = 45\). So \(x + 53=45\), then \(x=45 - 53=-8\). So the answer for that triangle is \(x=-8\) (option c).

Now, for the triangle type question (the blue triangle with angles \(39^\circ\), \(50^\circ\), \(91^\circ\)):

An obtuse triangle has one angle greater than \(90^\circ\). Since \(91^\circ>90^\circ\), the triangle is obtuse (option a).

For the triangle - angle matching question (question 44, but the text is a bit unclear, but let's assume it's about matching triangles with definitions):

• "all sides are the same length" is equilateral.
• "Has at least 2 equal sides" is isosceles.
• "Has no equal sides" is scalene.
• "Has one right angle" is right - angled.
• "Has 3 acute angles" is acute - angled.

But since the problem is a bit unclear in the image, we'll focus on the solved parts.

Answer:

For the triangle with angles \(70^\circ\), \(65^\circ\), \(x + 53\): \(x=-8\) (option c)

For the triangle type (angles \(39^\circ\), \(50^\circ\), \(91^\circ\)): obtuse (option a)

For the triangle with angles \(84^\circ\), \(x + 59\), \(x + 51\): \(x=-7\) (but if we take the first triangle's options, \(x=-8\) is option c)