{"id":3151,"date":"2023-05-19T19:15:24","date_gmt":"2023-05-19T19:15:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3151"},"modified":"2025-08-24T04:32:56","modified_gmt":"2025-08-24T04:32:56","slug":"introduction-to-geometry-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/introduction-to-geometry-learn-it-3\/","title":{"raw":"Introduction to Geometry: Learn It 3","rendered":"Introduction to Geometry: Learn It 3"},"content":{"raw":"

Supplementary and Complementary Angles<\/h2>\r\n

Are you familiar with the phrase 'do a [latex]180[\/latex]?' It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[\/latex] degrees.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"217\"]\"A Figure 1. The straight line has an angle of 180 degrees[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

We learned earlier in this section the basics of an angle. An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In the image below, [latex]\\angle A[\/latex] is the angle with vertex at point [latex]A[\/latex]. We measure angles in degrees, and use the symbol [latex]^ \\circ[\/latex] to represent degrees.
\"An<\/center>\r\n

 <\/p>\r\n<\/section>\r\n

The measure <\/strong>of [latex]\\angle A[\/latex] is written [latex]m\\angle A[\/latex]. So if [latex]\\angle A[\/latex] is [latex]\\text{27}^ \\circ [\/latex], we would write the measure as [latex]m\\angle A=\\text{27}^ \\circ[\/latex]. The measures of two angles can be added together. There are two special cases that occur when we add two angle measures: supplementary and complementary angles. If the sum of the measures of two angles is [latex]\\text{180}^ \\circ[\/latex], then they are called supplementary angles<\/strong>. In the images below, each pair of angles is supplementary because their measures add to [latex]\\text{180}^ \\circ [\/latex]. Each angle is the supplement<\/em> of the other.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"574\"]\"Part Figure 2. These angles are supplementary[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

If the sum of the measures of two angles is [latex]\\text{90}^ \\circ[\/latex], then the angles are complementary angles<\/strong>. In the images below, each pair of angles is complementary, because their measures add to [latex]\\text{90}^ \\circ[\/latex]. Each angle is the complement<\/em> of the other.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"286\"]\"Part Figure 3. These angles are complementary[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

\r\n
\r\n

supplementary and complementary angles<\/h3>\r\n\r\nIf the sum of the measures of two angles is [latex]\\text{180}^\\circ [\/latex], then the angles are supplementary<\/strong>.\r\n\r\n

If angle [latex]A[\/latex] and angle [latex]B[\/latex] are supplementary, then [latex]m\\angle{A}+m\\angle{B}=180^\\circ[\/latex].<\/p>\r\n

 <\/p>\r\n\r\nIf the sum of the measures of two angles is [latex]\\text{90}^\\circ[\/latex], then the angles are complementary<\/strong>.\r\n\r\n

If angle [latex]A[\/latex] and angle [latex]B[\/latex] are complementary, then [latex]m\\angle{A}+m\\angle{B}=90^\\circ[\/latex].<\/p>\r\n<\/div>\r\n<\/section>\r\n

In this section and the ones following, you will be introduced to some common geometry formulas. We will adapt our Problem-Solving Strategy for Geometry Applications to solve the problems you will encounter.\r\n\r\n

Problem-Solving Strategy for Geometry Applications<\/strong><\/p>\r\n

    \r\n\t
  1. Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\r\n\t
  2. Identify<\/strong> what you are looking for.<\/li>\r\n\t
  3. Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\r\n\t
  4. Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\r\n\t
  5. Solve<\/strong> the equation using good algebra techniques.<\/li>\r\n\t
  6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\r\n\t
  7. Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/section>\r\n

    The next example will show how you can use the Problem-Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.<\/p>\r\n

    An angle measures [latex]\\text{40}^ \\circ[\/latex].\r\n\r\n
      \r\n\t
    1. Find its supplement<\/li>\r\n\t
    2. Find its complement<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      1.<\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n
      \"Two<\/center>\r\n

       <\/p>\r\n<\/td>\r\n<\/tr>\r\n

      Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nThe supplement of a [latex]40\u00b0[\/latex]angle.<\/td>\r\n<\/tr>\r\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\nLet [latex]s=[\/latex]the measure of the supplement.<\/td>\r\n<\/tr>\r\n
      Step 4. Translate.<\/strong> Write the appropriate formula for the situation and substitute in the given information.<\/td>\r\n[latex]m\\angle A+m\\angle B=180[\/latex] [latex]s+40=180[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 5. Solve<\/strong> the equation.<\/td>\r\n[latex]s=140[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 6. Check:<\/strong> [latex]140+40\\stackrel{?}{=}180[\/latex] [latex]180=180\\checkmark[\/latex]<\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n
      Step 7. Answer<\/strong> the question.<\/td>\r\nThe supplement of the [latex]40\u00b0[\/latex]angle is [latex]140\u00b0[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
      2.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n
      \"Two<\/center>\r\n

       <\/p>\r\n<\/td>\r\n<\/tr>\r\n

      Step 2. Identify<\/strong> what you are looking for.<\/td>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      The complement of a [latex]40\u00b0[\/latex]angle.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\nLet [latex]c=[\/latex]the measure of the complement.<\/td>\r\n<\/tr>\r\n
      Step 4. Translate.<\/strong> Write the appropriate formula for the situation and substitute in the given information.<\/td>\r\n[latex]m\\angle A+m\\angle B=90[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 5. Solve<\/strong> the equation.<\/td>\r\n[latex]c+40=90[\/latex] [latex]c=50[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 6. Check:<\/strong> [latex]50+40\\stackrel{?}{=}90[\/latex] [latex]90=90\\quad\\checkmark[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
      Step 7. Answer<\/strong> the question.<\/td>\r\nThe complement of the [latex]40\u00b0[\/latex]angle is [latex]50\u00b0[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n
      [ohm2_question hide_question_numbers=1]7044[\/ohm2_question]<\/section>\r\n
      Two angles are supplementary. The larger angle is [latex]\\text{30}^ \\circ[\/latex] more than the smaller angle. Find the measure of both angles.[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer] [hidden-answer a=\"190834\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n
      \"Two<\/center>\r\n

       <\/p>\r\n<\/td>\r\n<\/tr>\r\n

      Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nThe measures of both angles.<\/td>\r\n<\/tr>\r\n
      Step 3. Name.<\/strong> Choose a variable to represent it. The larger angle is 30\u00b0 more than the smaller angle.<\/td>\r\nLet [latex]a=[\/latex] measure of smaller angle [latex]a+30=[\/latex] measure of larger angle<\/td>\r\n<\/tr>\r\n
      Step 4. Translate.<\/strong> Write the appropriate formula and substitute.<\/td>\r\n[latex]m\\angle A+m\\angle B=180[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 5. Solve<\/strong> the equation.<\/td>\r\n[latex](a+30)+a=180[\/latex] [latex]2a+30=180[\/latex] [latex]2a=150[\/latex] [latex]a=75=[\/latex] measure of smaller angle. [latex]a+30=[\/latex] measure of larger angle. [latex]75+30[\/latex] [latex]105[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 6. Check:<\/strong> [latex]m\\angle A+m\\angle B=180[\/latex] [latex]75+105\\stackrel{?}{=}180[\/latex] [latex]180=180\\quad\\checkmark[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
      Step 7. Answer<\/strong> the question.<\/td>\r\nThe measures of the angle are [latex]75\u00b0[\/latex]and [latex]105\u00b0[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\u00a0 [\/hidden-answer]<\/section>\r\n
      [ohm2_question hide_question_numbers=1]7047[\/ohm2_question]<\/section>\r\n
      You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property (see image below) and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to [latex]150^\\circ[\/latex]. What is the measure of the angle that will border your garden?
      \"Two<\/center>\r\n

       <\/p>\r\n\r\n[reveal-answer q=\"190839\"]Show Solution[\/reveal-answer] [hidden-answer a=\"190839\"] As the angle between Linton Street and First Avenue is [latex]150^\\circ[\/latex], the supplementary angle is [latex]30^\\circ[\/latex]. Therefore, the garden will form a [latex]30^\\circ[\/latex] angle at the corner of your property. [\/hidden-answer]<\/section>","rendered":"

      Supplementary and Complementary Angles<\/h2>\n

      Are you familiar with the phrase ‘do a [latex]180[\/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[\/latex] degrees.<\/p>\n

      \n
      \"A
      Figure 1. The straight line has an angle of 180 degrees<\/figcaption><\/figure>\n<\/div>\n

       <\/p>\n

      We learned earlier in this section the basics of an angle. An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In the image below, [latex]\\angle A[\/latex] is the angle with vertex at point [latex]A[\/latex]. We measure angles in degrees, and use the symbol [latex]^ \\circ[\/latex] to represent degrees.<\/p>\n
      \"An<\/div>\n

       <\/p>\n<\/section>\n

      The measure <\/strong>of [latex]\\angle A[\/latex] is written [latex]m\\angle A[\/latex]. So if [latex]\\angle A[\/latex] is [latex]\\text{27}^ \\circ[\/latex], we would write the measure as [latex]m\\angle A=\\text{27}^ \\circ[\/latex]. The measures of two angles can be added together. There are two special cases that occur when we add two angle measures: supplementary and complementary angles. If the sum of the measures of two angles is [latex]\\text{180}^ \\circ[\/latex], then they are called supplementary angles<\/strong>. In the images below, each pair of angles is supplementary because their measures add to [latex]\\text{180}^ \\circ[\/latex]. Each angle is the supplement<\/em> of the other.<\/p>\n

      \n
      \"Part
      Figure 2. These angles are supplementary<\/figcaption><\/figure>\n<\/div>\n

       <\/p>\n

      If the sum of the measures of two angles is [latex]\\text{90}^ \\circ[\/latex], then the angles are complementary angles<\/strong>. In the images below, each pair of angles is complementary, because their measures add to [latex]\\text{90}^ \\circ[\/latex]. Each angle is the complement<\/em> of the other.<\/p>\n

      \n
      \"Part
      Figure 3. These angles are complementary<\/figcaption><\/figure>\n<\/div>\n

       <\/p>\n

      \n
      \n

      supplementary and complementary angles<\/h3>\n

      If the sum of the measures of two angles is [latex]\\text{180}^\\circ[\/latex], then the angles are supplementary<\/strong>.<\/p>\n

      If angle [latex]A[\/latex] and angle [latex]B[\/latex] are supplementary, then [latex]m\\angle{A}+m\\angle{B}=180^\\circ[\/latex].<\/p>\n

       <\/p>\n

      If the sum of the measures of two angles is [latex]\\text{90}^\\circ[\/latex], then the angles are complementary<\/strong>.<\/p>\n

      If angle [latex]A[\/latex] and angle [latex]B[\/latex] are complementary, then [latex]m\\angle{A}+m\\angle{B}=90^\\circ[\/latex].<\/p>\n<\/div>\n<\/section>\n

      In this section and the ones following, you will be introduced to some common geometry formulas. We will adapt our Problem-Solving Strategy for Geometry Applications to solve the problems you will encounter.<\/p>\n

      Problem-Solving Strategy for Geometry Applications<\/strong><\/p>\n

        \n
      1. Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n
      2. Identify<\/strong> what you are looking for.<\/li>\n
      3. Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\n
      4. Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n
      5. Solve<\/strong> the equation using good algebra techniques.<\/li>\n
      6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n
      7. Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/section>\n

        The next example will show how you can use the Problem-Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.<\/p>\n

        An angle measures [latex]\\text{40}^ \\circ[\/latex].<\/p>\n
          \n
        1. Find its supplement<\/li>\n
        2. Find its complement<\/li>\n<\/ol>\n