{"id":3184,"date":"2023-05-22T17:30:45","date_gmt":"2023-05-22T17:30:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3184"},"modified":"2025-08-26T03:19:05","modified_gmt":"2025-08-26T03:19:05","slug":"introduction-to-geometry-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/introduction-to-geometry-learn-it-4\/","title":{"raw":"Introduction to Geometry: Learn It 4","rendered":"Introduction to Geometry: Learn It 4"},"content":{"raw":"

Vertical Angles<\/h2>\r\n

When two lines intersect, the opposite angles are called vertical angles<\/strong>, and vertical angles have equal measure. For example, the image below shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.<\/p>\r\n

\r\n[caption id=\"attachment_3221\" align=\"aligncenter\" width=\"351\"]\"Two Figure 1. These two intersecting lines have vertical angles[\/caption]\r\n<\/center>\r\n

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vertical angles<\/h3>\r\nVertical angles<\/strong> are a pair of opposite angles that are formed when two lines intersect. These angles are located across from each other and share a common vertex or point of intersection. Vertical angles have equal measures, which means they have the same angle measurement.<\/div>\r\n<\/section>\r\n
In the figure below, one angle measures [latex]40^\\circ[\/latex]. Find the measures of the remaining angles.
\"Two<\/center>\r\n

 <\/p>\r\n\r\n[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"] The 40-degree angle and [latex]\u22202[\/latex] are vertical angles. Therefore, [latex]m\u22202=40^\\circ[\/latex]. Notice that [latex]\u22202[\/latex] and [latex]\u22201[\/latex] are supplementary angles, meaning that the sum of [latex]m\u22202[\/latex] and [latex]m\u22201[\/latex] equals [latex]180^\\circ[\/latex]. Therefore, [latex]m\u22201=180^\\circ\u221240^\\circ=140^\\circ[\/latex]. Since [latex]\u22201[\/latex] and [latex]\u22203[\/latex] are vertical angles, then [latex]m\u22203[\/latex] equals [latex]140^\\circ[\/latex].[\/hidden-answer]<\/section>\r\n

[ohm2_question hide_question_numbers=1]7051[\/ohm2_question]<\/section>\r\n

Transversals<\/h2>\r\n

When two parallel lines are crossed by a straight line or transversal<\/strong>, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. In the image below, angles [latex]1[\/latex], [latex]2[\/latex], [latex]7[\/latex], and [latex]8[\/latex] are called exterior angles, and angles [latex]3[\/latex], [latex]4[\/latex], [latex]5[\/latex], and [latex]6[\/latex] are called interior angles.<\/p>\r\n

\r\n[caption id=\"attachment_3231\" align=\"aligncenter\" width=\"360\"]\"Two Figure 2. These lines have a transversal[\/caption]\r\n<\/center>\r\n

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Alternate interior angles<\/strong> are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example in the image below, [latex]\u22203[\/latex] and [latex]\u22206[\/latex] are alternate interior angles and have equal measure; [latex]\u22204[\/latex] and [latex]\u22205[\/latex] are alternate interior angles and have equal measure as well.<\/p>\r\n

\r\n[caption id=\"attachment_3232\" align=\"aligncenter\" width=\"360\"]\"Two Figure 3. Angles 3 and 6 are alternate interior angels[\/caption]\r\n<\/center>\r\n

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Alternate exterior angles<\/strong> are exterior angles on opposite sides of the transversal and have the same measure. For example in the image below, [latex]\u22202[\/latex] and [latex]\u22207[\/latex] are alternate exterior angles and have equal measures; [latex]\u22201[\/latex] and [latex]\u22208[\/latex] are alternate exterior angles and have equal measures as well.<\/p>\r\n

\r\n[caption id=\"attachment_3233\" align=\"aligncenter\" width=\"360\"]\"Two Figure 4. Angles 2 and 7 are alternate exterior angles[\/caption]\r\n<\/center>\r\n

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Corresponding angles<\/strong> refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In the image below, [latex]\u22201[\/latex] and [latex]\u22205[\/latex] are corresponding angles and have equal measures; [latex]\u22203[\/latex] and [latex]\u22207[\/latex] are corresponding angles and have equal measures; [latex]\u22202[\/latex] and [latex]\u22206[\/latex] are corresponding angles and have equal measures; [latex]\u22204[\/latex] and [latex]\u22208[\/latex] are corresponding angles and have equal measures as well.<\/p>\r\n

\r\n[caption id=\"attachment_3239\" align=\"aligncenter\" width=\"360\"]\"Two Figure 5. Angles 1 and 5 are corresponding angles[\/caption]\r\n<\/center>\r\n

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transversals, alternate interior angles, alternate exterior angles, and corresponding angles<\/h3>\r\n\r\nA transversal <\/strong>is a line that intersects two or more other lines, creating various angles. Alternate interior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the inside of the two other lines. These angles are equal in measure. Alternate exterior angles<\/strong> are a pair of angles that lie on opposite sides of the transversal and are on the outside of the two other lines. These angles are equal in measure. Corresponding angles<\/strong> are a pair of angles that are in the same relative position with respect to the transversal, but they are on different intersected lines. These angles are also equal in measure.<\/div>\r\n<\/section>\r\n
Using the figure below and given that angle [latex]3[\/latex] measures [latex]40^\\circ[\/latex], find the measures of the remaining angles and give a reason for your solution.
\"Two<\/center>\r\n

 <\/p>\r\n\r\n[reveal-answer q=\"57881\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57881\"] [latex]m\u22202=m\u22203=40^\\circ[\/latex] by vertical angles. [latex]m\u22207=m\u22203=40^\\circ[\/latex] by corresponding angles. [latex]m\u22207=m\u22206=40^\\circ[\/latex] by vertical angles. [latex]m\u22201=180\u221240=140^\\circ[\/latex] by supplementary angles. [latex]m\u22204=m\u22201=140^\\circ[\/latex] by vertical angles. [latex]m\u22208=m\u22201=140^\\circ[\/latex] by alternate exterior angles. [latex]m\u22205=m\u22208=140^\\circ[\/latex] by vertical angles. [\/hidden-answer]<\/section>\r\n

[ohm2_question hide_question_numbers=1]7052[\/ohm2_question]<\/section>","rendered":"

Vertical Angles<\/h2>\n

When two lines intersect, the opposite angles are called vertical angles<\/strong>, and vertical angles have equal measure. For example, the image below shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.<\/p>\n

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\"Two
Figure 1. These two intersecting lines have vertical angles<\/figcaption><\/figure>\n<\/div>\n

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vertical angles<\/h3>\n

Vertical angles<\/strong> are a pair of opposite angles that are formed when two lines intersect. These angles are located across from each other and share a common vertex or point of intersection. Vertical angles have equal measures, which means they have the same angle measurement.<\/div>\n<\/section>\n

In the figure below, one angle measures [latex]40^\\circ[\/latex]. Find the measures of the remaining angles.<\/p>\n
\"Two<\/div>\n

 <\/p>\n