Question

the difference between the measures of two complementary angles is \\(88^{\circ}\\). determine the measures of the two angles.

the larger angle has a measure of \\(\square^{\circ}\\).
the smaller angle has a measure of \\(\square^{\circ}\\).

Explanation:

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<reasoning>

Define the variables and system of equations

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &x + y = 90^\circ \\ &x - y = 88^\circ \end{aligned}$$

\]

Solve for the larger angle

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &(x + y) + (x - y) = 90^\circ + 88^\circ \\ &2x = 178^\circ \\ &x = 89^\circ \end{aligned}$$

\]

Solve for the smaller angle

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &89^\circ + y = 90^\circ \\ &y = 1^\circ \end{aligned}$$

\]
</reasoning>

<answer>
The difference between the measures of two complementary angles is \(88^\circ\). Determine the measures of the two angles.

The larger angle has a measure of <blank>89</blank>\(^\circ\).
The smaller angle has a measure of <blank>1</blank>\(^\circ\).
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Complementary Angles"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Define the variables and system of equations

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &x + y = 90^\circ \\ &x - y = 88^\circ \end{aligned}$$

\]

Solve for the larger angle

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &(x + y) + (x - y) = 90^\circ + 88^\circ \\ &2x = 178^\circ \\ &x = 89^\circ \end{aligned}$$

\]

Solve for the smaller angle

Using the Complementary Angles knowledge point
\[

$$\begin{aligned} &89^\circ + y = 90^\circ \\ &y = 1^\circ \end{aligned}$$

\]
</reasoning>

<answer>
The difference between the measures of two complementary angles is \(88^\circ\). Determine the measures of the two angles.

The larger angle has a measure of <blank>89</blank>\(^\circ\).
The smaller angle has a measure of <blank>1</blank>\(^\circ\).
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Complementary Angles"
]
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