Question

lines cv and rg intersect at point n are shown. which of the following statements are true? select all that apply. □ ∠cng and ∠rnv form a linear pair □ ∠rnc and ∠vng are supplementary angles □ m∠rnc + m∠cng + m∠gnv = 180° □ m∠rnv + m∠gnv = m∠rnc + m∠gnc □ ∠rnc and ∠vng are vertical angles

Explanation:

Step1: Recall angle - pair definitions

Linear - pair angles are adjacent angles that form a straight line and their sum is 180°. Vertical angles are non - adjacent angles formed by two intersecting lines and are equal. Supplementary angles are two angles whose sum is 180°.

Step2: Analyze each statement

• For the statement "∠CNG and ∠RNV form a linear pair":
• ∠CNG and ∠RNV are adjacent angles and they form a straight line at the intersection of lines CV and RG. So, the sum of ∠CNG and ∠RNV is 180°, and they form a linear pair.
• For the statement "∠RNC and ∠VNG are vertical angles":
• ∠RNC and ∠VNG are non - adjacent angles formed by the intersection of lines CV and RG. So, ∠RNC and ∠VNG are vertical angles.
• For the statement "m∠RNV + m∠GNV=m∠RNC + m∠GNC":
• Since ∠RNV and ∠RNC are vertical angles, and ∠GNV and ∠GNC are vertical angles, and vertical angles are equal, we have m∠RNV = m∠RNC and m∠GNV = m∠GNC. So, m∠RNV + m∠GNV=m∠RNC + m∠GNC.
• For the statement "m∠RNC + m∠CNG+m∠GNV = 180°":
• ∠RNC, ∠CNG, and ∠GNV together form a straight - line angle at point N on line RG. So, m∠RNC + m∠CNG+m∠GNV = 180°.
• For the statement "∠RNC and ∠VNG are supplementary angles":
• Since ∠RNC and ∠VNG are vertical angles, and vertical angles are equal, they are not supplementary (supplementary angles add up to 180° and vertical angles are equal in measure, not necessarily 90° each).

Answer:

∠CNG and ∠RNV form a linear pair
∠RNC and ∠VNG are vertical angles
m∠RNV + m∠GNV=m∠RNC + m∠GNC
m∠RNC + m∠CNG+m∠GNV = 180°